Chaotic Oscillation via Edge of Chaos Criteria

Transcription

1 International Journal of Bifurcation and Chaos, Vol. 27, No. 11 (2017) (79 pages) c World Scientific Publishing Company DOI: /S X Chaotic Oscillation via Edge of Chaos Criteria Makoto Itoh , Arae, Jonan-ku, Fukuoka , Japan itoh-makoto@jcom.home.ne.jp Leon Chua Department of Electrical Engineering and Computer Sciences, University of California, Berkeley, CA 94720, USA chua@berkeley.edu Received December 15, 2016; Revised September 22, 2017 In this paper, we show that nonlinear dynamical systems which satisfy the edge of chaos criteria can bifurcate from a stable equilibrium point regime to a chaotic regime by periodic forcing. That is, the edge of chaos criteria can be exploited to engineer a phase transition from ordered to chaotic behavior. The frequency of the periodic forcing can be derived from this criteria. In order to generate a periodic or a chaotic oscillation, we have to tune the amplitude of the periodic forcing. For example, we engineer chaotic oscillations in the generalized Duffing oscillator, the FitzHugh Nagumo model, the Hodgkin Huxley model, and the Morris Lecar model. Although forced oscillators can exhibit chaotic oscillations even if the edge of chaos criteria is not satisfied, our computer simulations show that forced oscillators satisfying the edge of chaos criteria can exhibit highly complex chaotic behaviors, such as folding loci, strong spiral dynamics, or tight compressing dynamics. In order to view these behaviors, we used high-dimensional Poincaré maps and coordinate transformations. We also show that interesting nonlinear dynamical systems can be synthesized by applying the edge of chaos criteria. They are globally stable without forcing, that is, all trajectories converge to an asymptotically-stable equilibrium point. However, if we apply a forcing signal, then the dynamical systems can oscillate chaotically. Furthermore, the average power delivered from the forced signal is not dissipated by chaotic oscillations, but on the contrary, energy can be generated via chaotic oscillations, powered by locally-active circuit elements inside the one-port circuit N connected across a current source. Keywords: Duffing oscillator; Hodgkin Huxley model; Morris Lecar model; FitzHugh Nagumo model; Van der Pol oscillator; edge of chaos criteria; average power; periodic forcing; chaos; local activity; sink; subcritical Hopf bifurcation. 1. Introduction In this paper, we show that nonlinear dynamical systems which satisfy the edge of chaos criteria [Chua, 2005, 2013] can bifurcate from a stable equilibrium point regime to a chaotic regime by periodic forcing. That is, the edge of chaos criteria can provide a phase transition from an ordered regime to a chaotic regime by tuning the amplitude of the periodic forcing. In particular, we will show that the generalized Duffing oscillator, the FitzHugh Nagumo model, the Hodgkin Huxley model, and the Morris Lecar model can exhibit chaotic oscillation by periodic forcing. Without loss of generality, we can choose parameters such that an equilibrium point can be chosen to be locally asymptotically stable without forcing. However, if we apply an appropriate periodic forcing signal which satisfies the edge of chaos criteria, then the periodic forcing

2 M. Itoh & L. Chua destabilizes the above equilibrium point, and the driven systems can give rise to a chaotic oscillation. We can use a sinusoidal signal, or a rectangularpulse, as the periodic forcing signals to generate a chaotic oscillation. We used the sinusoidal steady state linear circuit analysis to analyze the steady state oscillation and the stability of the equilibrium point. We emphasize that the frequency of the periodic forcing signal can be calculated by applying the edge of chaos criteria. In order to generate a periodic, or a chaotic oscillation, we have to tune the amplitude of the forcing signal. The above steps will be summarized as the chaos generation procedure in Sec. 7. Furthermore, we show that the above mentioned forced oscillators belong to a special group of forced nonlinear systems, which do not dissipate the average power associated with the forcing signal. However, the well-known Duffing oscillator and the forced Van der Pol oscillator are not included in this group. We also show that certain class of forced oscillators can be derived from the edge of chaos criteria. That is, they are globally stable without forcing where all trajectories converge to an asymptotically-stable equilibrium point. If we apply a forcing signal, then the forced dynamical systems can oscillate chaotically. Furthermore, the average power associated with the forced signal is not dissipated during chaotic oscillations, on the contrary energy is generated during chaotic oscillations. It is well-known that forced oscillators can exhibit chaotic oscillation, even if they do not satisfy the edge of chaos criteria. However, if the forced oscillators satisfy the edge of chaos criteria, then they can exhibit more complex chaotic behaviors, such as folding loci, strong spiral dynamics, tight compressing dynamics, etc. We used two methods to view these chaotic behaviors. One method analyzes the high-dimensional Poincaré mapsvia3d bubble charts. The other method applies a coordinate transformation to project the trajectories in the high-dimensional space into trajectories in a three-dimensional space. 2. Generalized Duffing Oscillator The Duffing oscillator is a forced oscillator defined by ẍ + kẋ + ax 3 + bx = γ cos ωt, (1) where k, a, b, γ, andω are constants. Equation (1) can be recast into a system of two first-order ordinary differential equations: ẋ = y, ẏ = ky ax 3 bx + γ cos(ωt). In this paper, we generalize Eq. (2) as follows: ẋ = y + cx, ẏ = ky ax 3 bx + j(t). (2) (3) We added the term cx to the first equation, and j(t) denotes a periodic forcing. The reason for adding cx in the first equation will be discussed in Sec. 3. Assume k =0.05, a =1, b =0.2, c =0.02. (4) Then Eq. (3) can be written as Generalized Duffing oscillator ẋ = y +0.02x, (5) ẏ = 0.05y x 3 0.2x + j(t). Equation (5) can be realized by the circuit in Fig. 1. Its dynamics can be given by L 1 di 1 dt = v 1 R 1 i 1, C 1 dv 1 dt = i 1 G 1 v 1 i i 1 + j(t) where = G 1 v 1 i i 1 + j(t), (6) C 1 =1, L 1 =1, G 1 =0.05, R 1 = (7) Substituting Eq. (7) into Eq. (6), we obtain di 1 dt = v i 1, (8) dv 1 dt = 0.05v 1 i i 1 + j(t). Equations (5) and (8) are identical, if we rename the variables i 1 = x, v 1 = y. (9)

3 Chaotic Oscillation via Edge of Chaos Criteria Fig. 1. Circuit model for Eq. (5). Parameters: C 1 =1,L 1 =1,G 1 =0.05, R 1 = The nonlinear current-controlled current source is defined by i c = i i 1. The independent current source j(t) denotes a periodic forcing Equilibrium point Let us study the equilibrium point (that is, the solution which does not change with time) of the following autonomous system 1 ẋ = y +0.02x, ẏ = 0.05y x 3 0.2x, (10) obtained by setting j(t) = 0 in Eq. (5). Note that Eq. (10) does not have the forcing term j(t). It has only one equilibrium point at the origin: (x, y) = (0, 0). The Jacobian matrix J of the linearized system at this point is given by [ ] J =. (11) The characteristic equation associated with J is: λ λ = 0. (12) Its eigenvalues are given by λ = 0.03 ± i ± i (13) Thus, the equilibrium point of Eq. (10), namely, the origin, is locally asymptotically stable. We next show that the above equilibrium point is globally asymptotically stable. Let X(x, y) be the vector field of Eq. (10), that is, X(x, y) =(ẋ, ẏ) Then, we obtain =(y +0.02x, 0.05y x 3 0.2x). (14) div X(x, y) = (15) Thus, Eq. (10) has no closed trajectories by Bendixson s criterion [Andronov et al., 1987]. Since Eq. (10) has only one asymptotically-stable equilibrium point at the origin, this equilibrium point becomes globally asymptotically stable Variational equation Let δx and δy denote infinitesimal variables in the neighborhood of the equilibrium point (x 0,y 0 ) = (0, 0) of Eq. (5), namely, x(t) =x 0 + δx(t) =δx(t), y(t) =y 0 + δy(t) =δy(t). (16) From Eq. (5), we obtain the variational equation at the origin: Variational equation d(δx) dt d(δy) dt = δy +0.02δx, = 0.05δy 0.2δx + j(t). (17) 1 A system of ordinary differential equations is said to be autonomous if it does not explicitly contain time t [Guckenheimer & Holmes, 1983]

4 M. Itoh & L. Chua 2.3. Laplace transform of the variational equation Let us define the Laplace transform of δx, δy, and j(t) as follow: Definition of Laplace transform ˆx(s) = ŷ(s) = ĵ(s) = δx(t)e st dt, δy(t)e st dt, j(t)e st dt, (18) where σ + iω C. (s 0.02)δy(0) 0.2δx(0) G(s) s 2. (26) +0.03s Applying the Laplace transform to each term The function G(s) has two poles: in Eq. (17), we obtain sˆx(s) δx(0) = ŷ(s)+ 0.02ˆx(s), p 1 = i i0.4458, (19) 2 sŷ(s) δy(0) = 0.05ŷ(s) 0.2ˆx(s)+ĵ(s), p where δx(0) and δy(0) denote the initial condition. 2 = 0.03 i i Solving the first equation in Eq. (19) for ˆx(s), (27) we obtain Thus, G(s) can be written as ˆx(s) =ŷ(s)+δx(0). (20) s 0.02 The second equation in Eq. (19) can be written as (s )ŷ(s)+ 0.2ˆx(s) = δy(0) + ĵ(s). (21) Substituting Eq. (20) into Eq. (21), we obtain (ŷ(s)+δx(0) ) (s +0.05)ŷ(s)+0.2 s 0.02 = δy(0) + ĵ(s). (22) The left-hand side can be written as { } (s +0.05)(s 0.02) ŷ(s)+ 0.2δx(0) s 0.02 s 0.02 ( s 2 ) +0.03s = ŷ(s)+ 0.2δx(0) s 0.02 s (23) Thus, Eq. (22) can be written as ( s 2 ) +0.03s ŷ(s) s 0.02 = ĵ(s)+δy(0) 0.2δx(0) s (24) From Eq. (24), we obtain Laplace transform of δy(t) ŷ(s) = s 0.02 s s ĵ(s) + (s 0.02)δy(0) 0.2δx(0) s 2. (25) +0.03s Transient term Let us study the transient term of Eq. (25), that is, the second equation of the left-hand side of this equation; namely, G(s) = (s 0.02)δy(0) 0.2δx(0) (s p 1 )(s p 2 ) = K a + K b, (28) s p 1 s p 2 where K a and K b is calculated by K a =(s p 1 )G(s) s=p1 = (p )δy(0) 0.2δx(0) p 1 p 2, K b =(s p 2 )G(s) s=p2 = (p )δy(0) 0.2δx(0) p 2 p 1. From Eq. (28), we obtain (29) g(t) L 1 [G(s)] = K a e p 1t + K a e p 2t 0, (30) as t, where the symbol L 1 indicates the inverse Laplace transform. Note that p 1 and p 2 have a negative real part. In order to study the steady state of y(t), we can assume that δx(0) = 0, δy(0) = 0, that is, zero

5 Chaotic Oscillation via Edge of Chaos Criteria initial condition. From Eqs. (25) and (26), we obtain Transient term of ŷ(s) G(s) = Thus, we obtain (s 0.02)δy(0) 0.2δx(0) s s (31) Laplace transforms ŷ(s) andˆx(s) ŷ(s) = ˆx(s) = = s 0.02 s s ĵ(s), ŷ(s) s s s ĵ(s). Furthermore, we obtain from Eq. (16) x(0) = x 0 + δx(0) = x 0 =0, y(0) = y 0 + δy(0) = y 0 =0, (32) (33) where we assumed that δx(0) = 0,δy(0) = 0. Thus, the initial condition for the Laplace transform is at the equilibrium point (origin). From Eq. (32), we can define the impedance Z(s) at the equilibrium point (0, 0) of the one-port N by Impedance Z(s) Z(s) ŷ(s) ĵ(s) = s 0.02 s s (34) We show the circuit realization of Z(s) in Sec The impedance Z(s) has two poles: p 1 = i p 2 = 0.03 i i0.4458, i (35) Since they are in the open left-half plane, our steady state analysis confirms our earlier time-domain analysis that the equilibrium point of Eq. (10) is locally asymptotically stable. Fig. 2. Small-signal circuit model for Eq. (17) (left). This one-port N has an impedance defined by Z(s) L[v(t)] L[j = 1,wherethesymbolN indicates s(t)] 1 sc + G + sl + R the Laplace transform. The periodic forcing j s(t) is supplied by connecting the independent current source (right) across the one-port N Small-signal circuit model Let us study the small-signal circuit model for Eq. (17). From Eq. (22), Z(s) can be recast into 1 Z(s) = s s =. (36) 1 s s 0.1 The impedance Z(s) of the linear circuit in Fig. 2 is given by Z(s) = L[v(t)] L[j s (t)] = 1, (37) 1 sc + G + sl + R where v(t) denotes the voltage across the capacitor C and j s (t) denotes the independent current source. Thus, Eq. (36) can be realized by this circuit. Its parameters are given by C =1, L =5, G =0.05, R = 0.1. (38) Observe that this circuit has a locally-active element R with a negative resistance equal to 0.1Ω and Z(s) =Z(s). If we apply the periodic forcing j s (t) to the small-signal circuit model in Fig. 2, then the dynamics of this circuit is given by L di dt = v Ri, C dv dt = i Gv + j s(t), (39)

6 M. Itoh & L. Chua that is, di =0.2v +0.02i, dt dv dt = i 0.05v + j s(t). (40) Applying the Laplace transform to each term in Eq. (40) under zero initial condition, we can obtain Eqs. (34) and (36). Here, we choose an alternative way, that is, we obtain Eq. (34) using the relationship among the state variables. If we set v =5 v and j s (t) =5j(t), then we obtain from Eq. (40) di = v +0.02i, dt (41) d v = 0.2i 0.05 v + j(t). dt It is equivalent to Eq. (17). That is, we obtain the relationship Hence, i(t) =δx(t), v(t) =δy(t). (42) L[v(t)] L[j s (t)] = L[5 v(t)] L[5j(t)] = L[ v(t)] L[j(t)] = L[δy(t)] L[j(t)] = ŷ(s) ĵ(s) where L denotes the Laplace transform. = Z(s), (43) 2.6. Sinusoidal steady state of δy(t) If we assume that the periodic forcing is given by a sinusoidal current source, 2 that is, then we obtain j(t) =A sin(ωt), (44) ĵ(s) =L[j(t)] = L[A sin(ωt)] = Furthermore, if we assume then ŷ(s) is given by Aω s 2 + ω 2. (45) δx(0) = 0, δy(0) = 0, (46) Aω ŷ(s) =Z(s)ĵ(s) =Z(s) s 2 + ω 2 = AωZ(s) s 2 + ω 2. (47) Substituting Eq. (34) into Eq. (47), we obtain ŷ(s) = AωZ(s) s 2 + ω 2 ( )( ) Aω s 0.02 = s 2 + ω 2 s s = Aω(s 0.02) (s + iω)(s iω)(s p 1 )(s p 2 ), (48) where p 1 and p 2 are given in Eq. (35). Hence, ŷ(s) canbeexpressedas Partial fraction form of ŷ(s) where ŷ(s) = K a s + iω + K a s iω + K 1 + K 2, s p 1 s p 2 K a = ( ) AωZ(s) s 2 + ω 2 (s iω) ωaz (iω) = = AZ(iω), 2iω 2i ( ) AωZ(s) K a = s 2 + ω 2 (s + iω) s=iω s= iω ωaz ( iω) = = AZ( iω), 2iω 2i ( ) AωZ(s) K 1 = s 2 + ω 2 (s p 1 ), s=p1 ( ) AωZ(s) K 2 = s 2 + ω 2 (s p 2 ), s=p2 and Z(iω) can be written as (49) (50) iω 0.02 Z(iω) = ω i0.03ω [ 0.01(5ω 2 ] 0.398) = (0.199 ω 2 ) 2 +(0.03ω) 2 [ ω ω 3 ] + i (0.199 ω 2 ) 2 +(0.03ω) 2, (51) 2 We use the sine function in order to apply the standard steady state analysis method [Van Valkenburg, 1964] to Eq. (34)

7 Chaotic Oscillation via Edge of Chaos Criteria and Real part and imaginary part of Z(iω) [ 0.01(5ω ) Re[Z(iω)] = (0.199 ω 2 ) 2 +(0.03ω) 2 [ ω ω 3 Im[Z(iω)] = (0.199 ω 2 ) 2 +(0.03ω) 2 Furthermore, Z(iω) satisfies the relation ], ]. The inverse Laplace transform of ŷ(s) becomes L 1 [ŷ(s)] = δy(t) =A Z(iω) sin(ωt + θ(ω)) + K 1 e p1t + K 2 e p2t, (58) (52) since [ ] L 1 e iθ(ω) s iω e iθ(ω) s + iω = e i(θ(ω)+ωt) e i(θ(ω)+ωt) If we define θ(ω) by that is, Z(iω) = Z( iω). (53) θ(ω) =argz(iω), (54) Z(iω) = Z(iω) e i{arg Z(iω)} = Z(iω) e iθ(ω), then we obtain (55) arg Z(iω) = arg Z( iω) = θ(ω), (56) (for more details see [Van Valkenburg, 1964]). Thus, ŷ(s) can be written as ŷ(s) = K a s + iω + = AZ(iω) 2i K a s iω + K 1 + K 2 s p 1 s p 2 ( ) 1 + AZ( iω) s iω 2i + K 1 + K 2 s p 1 s p 2 = A Z(iω) eiθ(ω) 2i + A Z(iω) e iθ(ω) 2i + K 1 + K 2 s p 1 s p 2 ( = A Z(iω) 2i ( 1 s iω ) ( 1 s + iω ) e iθ(ω) s iω e iθ(ω) s + iω ) ( 1 s + iω + K 1 s p 1 + K 2 s p 2. (57) ) =2isin(ωt + θ(ω)), [ ] L 1 K1 = K 1 e p1t, s p 1 [ ] L 1 K2 = K 2 e p2t. s p 2 From Eq. (58), we obtain Steady state of δy(t) δy(t) =y(t) y 0 = y(t) A Z(iω) sin(ωt + θ(ω)), for t. (60) (59) Note that p 1 and p 2 have a negative real part and y 0 = Sinusoidal steady state of δx(t) Let us study the behavior of δx(t) of the variational equation (17) for t. As stated before, we can assume that δx(0) = 0 and δy(0) = 0. From Eq. (32), ˆx(s) isgivenby Laplace transform of ˆx(s) where ( Aω ˆx(s) =Z 2 (s)ĵ(s) =Z 2 (s) Z 2 (s) = ĵ(s) = s 2 + ω 2 1 s s , Aω s 2 + ω 2. ), (61) (62)

8 M. Itoh & L. Chua Here, the periodic forcing is given by j(t) =A sin(ωt). The inverse Laplace transform of ˆx(s), namely, δx(t) satisfies the following relation: Steady state of δx(t) δx(t) =x(t) x 0 = x(t) A Z 2 (iω) sin(ωt + θ 2 (ω)), for t. (63) Fig. 3. Trajectories of the variational equation (17) with the periodic forcing j(t) = 5.5sin(0.2t). All trajectories tend to a Here, x 0 =0andZ 2 (iω) andθ 2 (ω) aregivenby periodic orbit (red). A transient trajectory is shown in black. Initial condition: (δx(0),δy(0)) = (0, 0). ( ) 1 Z 2 (iω) = ω 2, + i0.03ω (64) criteria of the edge of chaos [Chua, 2005, 2013]: θ 2 (ω) =argz 2 (iω). Edge of Chaos Criteria 2.8. Steady state oscillation Consider Eqs. (48) and (61). If A = 0, then we obtain ˆx(s) =ŷ(s) = 0, that is, x(t) =y(t) =0. Thus, the trajectory does not move from the asymptotically-stable equilibrium point. However, if we apply the periodic forcing j(t) witha 0, then the trajectory of Eq. (17) tends to the periodic orbit 3 on the (δx, δy)-plane, which is defined by Periodic orbit δx(τ) =A Z 2 (iω) sin(ωτ + θ 2 (ω)), δy(τ) = A Z(iω) sin(ωτ + θ(ω)), where 0 τ 2π ω. (65) Observe that the amplitude of the periodic orbits grows as A increases. This behavior is similar to a Hopf bifurcation. We show the above periodic orbit of Eq. (3) in Fig. 3, where the periodic forcing is given by j(t) = 5.5sin(0.2t), where we set ω =0.2. Let us next show that the above steady state oscillation is closely related to the mathematical From our steady state analysis, we conclude as follows: Condition (i) All poles of Z(s) are on the open left-half plane. Condition (ii) Re[Z(iω)] < 0 for at least one frequency ω = ω 0. (1) The sinusoidal forcing j(t) in Eq. (17) destabilizes the equilibrium point of Eq. (10), and causes a steady state oscillation in Eq. (17). (2) If the edge of chaos condition (i) is not satisfied, then Eq. (17) does not exhibit a steady state oscillation. We note that the above results are valid for Eq. (5) only in an infinitesimal neighborhood of the equilibrium point of Eq. (10). That is, the generalized Duffing oscillator (5) also exhibits a steady state oscillation, if A is sufficiently small. Observe that the trajectory of Eq. (5) tends to a periodic orbit. 3 3 Without loss of generality, we can use the terminology periodic orbit in order to describe a periodic trajectory ofthe nonautonomous systems, such as the generalized Duffing oscillator (5) and the variational equation (17) (see Duffing s Equation in Sec. 2.2 of [Guckenheimer & Holmes, 1983])

9 Chaotic Oscillation via Edge of Chaos Criteria 2.9. Real part of Z(iω) Let us next examine the real part of Z(iω) in Eq. (52): [ 0.01(5ω 2 ] 0.398) Re[Z(iω)] = (0.199 ω 2 ) 2 +(0.03ω) 2. (66) We found that Re[Z(iω)] < 0for0 ω 2 < 0.398, 5 that is, ω< (67) Fig. 4. Trajectories of the generalized Duffing oscillator defined by Eq. (5) with j(t) = 0.01 sin(0.2t). In this case, the amplitude of the periodic forcing is very small, that is, A trajectory tends to a periodic orbit (green). A transient trajectory is shown in black. Initial condition: (x(0),y(0)) = (0, 0). The amplitude of the periodic orbits grows as A increases, and becomes increasingly more complicated as shown in Fig. 5. When A increases further, Eq. (5) exhibits more complicated chaotic oscillations as will be shown in Sec We cannot explain this kind of complexity by using the edge of chaos criteria. This is because the edge of chaos criteria is derived from a linearized system. We show the frequency response of the impedance Re[Z(iω)] in Fig. 6. Thus, if ω satisfy Eq. (67), then the edge of chaos criteria is satisfied: Condition (i) All poles of Z(s) are on the open left-half plane. Condition (ii) Re[Z(iω)] < 0 for at least one frequency ω = ω 0. If the edge of chaos condition (i) is satisfied, that is, if all poles of Z(s) are on the open left-half plane, then the equilibrium point (origin) of Eq. (10) is locally asymptotically stable. The physical meaning of the edge of chaos condition (ii) is given in Sec Average power Since Z(s) is defined as an impedance in Eq. (34), ĵ(s) andŷ(s) are assumed to be the current and the voltage, respectively [see Eqs. (9), (34), (37), (42), and (43)]. Thus, we can define the instantaneous power entering the one-port N by 4 Fig. 5. Trajectories of the generalized Duffing oscillator defined by Eq. (5) with j(t) =0.5sin(0.2t). In this case, the amplitude of the periodic forcing is not so small, that is, 0.5. A trajectory tends to a slightly more complicated periodic orbit (blue). A transient trajectory is shown in black. Initial condition: (x(0),y(0)) = (0, 0). p(t) = δy(t)j(t). (68) If we choose ω = ω 0 satisfying Re[Z(iω 0 )] < 0, then after a sufficiently large number n of periods T of j(t) when the transient component is negligible, 4 Here, we assume j(t) =A sin (ωt) where the amplitude A is sufficiently small for our small-signal linear analysis in Sec. 2.9 to be applicable

10 M. Itoh & L. Chua Fig. 6. Small-signal impedance Z(iω) frequency response and its partially enlarged view (right). We found that Re[Z(iω)] < 0 for 0 ω 2 < r 0.398,thatis,0 ω< we obtain P n = 1 T = 1 T 1 T = 1 T = 1 T nt (n 1)T nt (n 1)T nt (n 1)T p(τ)dτ δy(τ)j(τ)dτ {A sin(ω 0 τ)}dτ nt (n 1)T {A Z(iω 0 ) sin(ω 0 τ + θ(ω 0 ))} A 2 Z(iω 0 ) {sin(ω 0 τ + θ(ω 0 )) sin(ω 0 τ)}dτ nt (n 1)T A 2 2 Z(iω 0) {cos θ(ω 0 ) cos(2ω 0 τ + θ(ω 0 ))}dτ = A2 2 Z(iω 0) cos θ(ω 0 ) = A2 2 Re[Z(iω 0)] < 0, (69) where j(t) =A sin(ω 0 t), T = 2π ω 0,andn is a sufficiently large integer. Thus, we obtain ( P (N) = 1 N ) P n < 0, (70) N n=1 for sufficiently large N. That is, the average power P (N) entering the one-port N is negative, if we apply the periodic forcing j(t) =A sin(ω 0 t)andn is sufficiently large. In this case, the small-signal circuit model defined by Z(s) is (locally) active, that is, it generates energy. Since P (N) can be recast into { P (N) = 1 N } P n N = 1 N = 1 NT = 1 NT n=1 { N ( )} 1 nt p(τ)dτ T n=1 (n 1)T { N ( )} nt p(τ)dτ n=1 { NT 0 (n 1)T } p(τ)dτ, (71) we can define the following continuous time average power Average power P (t) = 1 t = 1 t t 0 t 0 p(τ)dτ δy(τ)j(τ)dτ. (72) If we choose ω 0 satisfying Re[Z(iω 0 )] < 0, then we obtain P (t) < 0 for sufficiently large t as shown in Fig. 7(a). Here, we set ω 0 = 0.2, and theperiodicforcingisgivenbyj(t) =5.5sin(0.2t). In contrast, Fig. 7(b) shows P (t) > 0 for all t>0forω 1 =0.3 >ω 0. Here, ω 1 =0.3 satisfies Re[Z(iω 1 )] > 0, and the periodic forcing is given by j(t) =5.5sin(0.3t)

11 Chaotic Oscillation via Edge of Chaos Criteria (a) (b) Fig. 7. Average power P (t) calculated from the variational equation (17). (a) P (t) < 0fort 0, if the angular frequency ω 0 satisfies Re[Z(iω 0 )] < 0. Here, we choose ω 0 =0.2, and the periodic forcing of Eq. (17) is given by j(t) =5.5sin(0.2t). Initial condition: (δx(0),δy(0)) = (0, 0). (b) P (t) > 0fort>0, if the angular frequency ω 1 satisfies Re[Z(iω 1 )] > 0. Here, we choose ω 1 =0.3, and the periodic forcing of Eq. (17) is given by j(t) =5.5sin(0.3t). Initial condition: (δx(0),δy(0)) = (0, 0). Let us define the locally-active angular frequency as follows: Locally-active angular frequency Let Z(s) denotetheimpedance of a small-signal circuit model. Assume that it is driven by a current source j(t) =A sin (ω 0 t). Then the circuit is said to be forced at a locally-active angular frequency ω 0 if Re[Z(iω 0 )] < 0. Then, we conclude as follows: (1) The edge of chaos condition (ii) requires that the small-signal circuit model has a locally-active angular frequency ω 0,which satisfies Re[Z(iω 0 )] < 0. (2) Assume Re[Z(iω 0 )] < 0. Then, the corresponding small-signal circuit model must have at least one active component which can generate energy. (3) Assume Re[Z(iω 0 )] < 0 and assume that the corresponding small-signal circuit model is forced by j(t) =A sin(ω 0 t). Then the average power P (t) entering the one-port N becomes negative for sufficiently large t Computer simulations Let us first study the case where Eq. (5) is unforced, that is, j(t) = 0. Then the generalized Duffing oscillator (5) is an autonomous system. We show the trajectory of Eq. (5) in Fig. 8(a). It tends to a glob- asymptotically-stable equilibrium point (origin) ally as t. 5 We next study the case where Eq. (5) is forced by the periodic signal j(t) 0. That is, the system is a nonautonomous system. Then the periodic forcing destabilizes the equilibrium point (origin) of the unforced autonomous system, and a stable peri- oscillation appears. It can evolve to a odic chaotic oscillation as the amplitude A of the periodic forcing is increased. We show the chaotic trajectories of Eq. (5) in Figs. 8(b) and 8(c), where the periodic forcing is given by and j(t) =A sin(ωt), (73) j(t) =A s[sin(ωt)], (74) where A = 6, ω = 0.2, and s[z] denotes the unit step function, equal to 0 for z<0and1forz 0. Thus, Eq. (74) represents a rectangular-pulse, which can be used to study the pulse response. Note that we can calculate via explicit formula, the angular frequency ω which satisfies the edge of chaos criteria. However, we have to find the value of A by brute-force simulations until Eq. (5) exhibits chaos, because the edge of chaos criteria is valid only for a small neighborhood of the equilibrium point (origin). In order to view the trajectories in Fig. 8 from a different perspective, let us project the trajectories 5 Equilibrium point is defined only for autonomous systems. When j(t) = 0, Eq. (5) is an autonomous system. Thus, we used the terminology equilibrium point

12 M. Itoh & L. Chua (a) j(t) =0 (b)j(t) =6sin(0.2t) (c) j(t) =6s[sin(0.2t)] Fig. 8. Trajectories of the generalized Duffing oscillator defined by Eq. (5). We set ω 0 =0.2, which satisfies Re[Z(iω 0 )] < 0. (a) A trajectory of Eq. (5) tends to the asymptotically-stable equilibrium point (the origin) as t if j(t) = 0. In this case, Eq. (5) is an autonomous system. Initial condition: x(0) = 0.5,y(0) = 0.5. (b) Equation (5) exhibits a chaotic behavior when driven by j(t) = 6 sin(0.2t). Initial condition: x(0) = 0.5,y(0) = 0.5. (c) Equation (5) exhibits an interesting chaotic-like trajectory when driven by a rectangular-pulse forcing function j(t) =6s[sin(0.2t)], where s(u) denotes the unit step function, equal to 0 for u < 0 and 1 for u 0. Initial condition: x(0) = 0.5,y(0) = 0.5. into the (ξ,η,ζ)-space via the transformation ξ(t) =(x(t)+5)cos(0.2t), η(t) =(x(t)+5)sin(0.2t), ζ(t) =y(t). (75) For example, if (x(t),y(t)) moves on the unit circle (x(t),y(t)) = (cos(3t), sin(3t)), (76) then the projected orbit (ξ(t),η(t),ζ(t)) moves on a torus. That is, if we substitute Eq. (76) into Eq. (75), it gives a closed path on a torus (for more details, see Appendix A). Figure 9 shows the trajectories of Eq. (5), which are projected into the (ξ,η,ζ)-space. That is, the trajectory which tends to an asymptotically-stable equilibrium point (origin) on the (x, y)-plane is transformed into the trajectory which tends to a green circle in the (ξ,η,ζ)-space, as shown in Fig. 9(a). The chaotic attractor on the (x, y)-plane is transformed into the chaotic attractor in the three-dimensional (ξ,η,ζ)- space, as shown in Figs. 9(b) and 9(c). Observe the difference among the three orbits. The attractors in Figs. 9(b) and 9(c) look like a cruller donut. We also show their Poincaré maps in Fig Observe that their attractors are quite different. The attractor in 6 If we plot the intersection of the points with the plane defined by {(ξ,η, ζ) R 3 η =0,ξ 0}, we obtain similar Poincaré maps

13 Chaotic Oscillation via Edge of Chaos Criteria (a) j(t) =0 (c) j(t) =6s[sin(0.2t)] (b)j(t) =6sin(0.2t) Fig. 9. Projection of the trajectories of Eq. (5) into the (ξ,η,ζ)-space via the coordinate transformation (75). We set ω 0 =0.2, which satisfies Re[Z(iω 0 )] < 0. Observe the difference among the three orbits. The attractor in Fig. 9(b) looks like a twisted cruller donut. The attractor in Fig. 9(c) looks like a French cruller donut (typical round shape). The trajectory is coded using the rainbow color code for ζ, that is, the color evolves through violet, indigo, blue, green, yellow, orange and red, as ζ varies from its minimum to its maximum value. (a) A trajectory for j(t) = 0, which converges to a thick green circle with a radius of 5 after the transient regime (printed in rainbow color sequence). Since the trajectory on the (x, y)-plane tends to an asymptotically-stable equilibrium point (origin) as shown in Fig. 8(a), the corresponding trajectory in the (ξ,η, ζ)-space tends to a green circle. That is, the equilibrium point (origin) of Eq. (5) with j(t) = 0 is transformed into a green circle by the coordinate transformation (75) (see Appendix A for more details). Initial condition: x(0) = 0.5,y(0) = 0.5. The initial point (x(0),y(0)) = (0.5, 0.5) for Eq. (5) is projected into the point (ξ(0),η(0),ζ(0)) = (5.5, 0, 0.5) via the coordinate transformation (75). (b) A chaotic trajectory under the sinusoidal forcing j(t) = 6sin(0.2t). Since Eq. (5) exhibits a chaotic oscillation on the (x, y)-plane as shown in Fig. 8(b), the corresponding trajectory in the (ξ,η,ζ)-space is also chaotic (see Appendix A for more details). Initial condition: x(0) = 0.5,y(0) = 0.5. (c) A chaotic trajectory under the rectangular-pulse forcing j(t) = 6 s[sin(0.2t)]. Since Eq. (5) exhibits a chaotic-like oscillation on the (x, y)-plane as shown in Fig. 8(c), the corresponding trajectory in the (ξ,η, ζ)-space looks chaotic (see Appendix A for more details). Initial condition: x(0) = 0.5,y(0) =

14 M. Itoh & L. Chua Periodic solution in the generalized Duffing oscillator (5) For small periodic forcing, Eq. (5) exhibits a small periodic orbit (periodic response). If we increase the amplitude of the periodic forcing further, the periodic orbit (periodic response) grows, and evolves into a chaotic trajectory if ω is a locally-active angular frequency. (a) j(t) =6sin(0.2t) (b) j(t) =6s[sin(0.2t)] Fig. 10. Poincaré maps of the generalized Duffing oscillator defined by Eq. (5). We set ω 0 = 0.2, which satisfies Re[Z(iω 0 )] < 0. (a) Poincaré map under the sinusoidal forcing j(t) = 6sin(0.2t). The attractor exhibits stretch-andfold dynamics. Initial condition: x(0) = 0.5,y(0) = 0.5. (b) Poincaré map under the rectangular-pulse forcing j(t) = 6 s[sin(0.2t)]. The attractor exhibits spiral dynamics. Initial condition: x(0) = 0.5,y(0) = 0.5. Fig. 10(a) exhibits stretch-and-fold dynamics, and the attractor in Fig. 10(b) exhibits spiral dynamics. From our computer simulations, we obtain the following result: We show the chaotic attractors in Figs. 11 and 12 for some angular frequency ω 0, which satisfies Re[Z(iω 0 )] < 0. Observe the following behavior of chaotic attractors 7 : Behavior of attractors for Re[Z(iω 0 )] < 0 Sinusoidal forcing The chaotic attractor is increasingly compressed until it looks like a one-dimensional curve as Re[Z(iω 0 )] < 0 decreases. Rectangular-pulse forcing The chaotic spiral attractor tightens and the spiral s arm grows gradually as Re[Z(iω 0 )] < 0 decreases. Observe that y(t) and j(t) are equivalent to the voltage v 1 (t) across the capacitor C 1 and the current j(t) of the independent current source, respectively [see Fig. 1 and Eqs. (6) and (9)]. That is, y(t) and j(t) are equivalent to the terminal voltage and current of the independent current source in Fig. 1, respectively. Thus, we can define the instantaneous power entering the one-port N at time τ by y(τ)j(τ), and the average power P (t) by 8 P (t) = 1 t t 0 y(τ)j(τ)dτ. (77) We show the average power P (t) of the generalized Duffing oscillator in Fig. 13. The periodic forcing signals are given by j(τ) =6sin(0.2τ), (78) 7 The chaotic attractors of Eq. (5) have one positive Lyapunov exponent and one negative Lyapunov exponent. We could not calculate the Lyapunov exponents of chaotic attractors precisely, in the neighborhood of the phase transition region. However, we can observe the difference among chaotic attractors by plotting Poincaré maps and applying appropriate coordinate transformations. 8 Note that Re[Z(iω 0 )] < 0 does not always mean P (t) < 0 for sufficiently large t, since the generalized Duffing oscillator is not a linear dynamical system

15 Chaotic Oscillation via Edge of Chaos Criteria (a) ω 0 =0.23, Re[Z(iω 0 )] (b) ω 0 =0.2, Re[Z(iω 0 )] (c) ω 0 =0.1, Re[Z(iω 0 )] (d) ω 0 =0.095, Re[Z(iω 0 )] Fig. 11. Attractors of the generalized Duffing oscillator (5) with decreasing Re[Z(iω 0 )] from (a) to (d). The angular frequency ω 0 of the sinusoidal forcing satisfies Re[Z(iω 0 )] < 0. Observe that the distance between neighboring branches of the chaotic attractor is increasingly compressed until it looks like a one-dimensional curve as Re[Z(iω 0 )] decreases (see also Sec. 6 of [Thompson & Stewart, 1986]). (a) Sinusoidal forcing j(t) =6sin(0.23t). Re[Z(iω 0 )] for ω 0 =0.23. Initial condition: x(0) = 0.5,y(0) = 0.5. (b) Sinusoidal forcing j(t) =6sin(0.2t). Re[Z(iω 0 )] for ω 0 =0.2. Initial condition: x(0) = 0.5,y(0) = 0.5. (c) Sinusoidal forcing j(t) =6sin(0.1t). Re[Z(iω 0 )] for ω 0 =0.1. Initial condition: x(0) = 0.5,y(0) = 0.5. (d) Sinusoidal forcing j(t) =6sin(0.095t). Re[Z(iω 0 )] for ω 0 = Initial condition: x(0) = 0.5,y(0) =

16 M. Itoh & L. Chua (a) ω 0 =0.245, Re[Z(iω 0 )] (b) ω 0 =0.2, Re[Z(iω 0 )] (c) ω 0 =0.15, Re[Z(iω 0 )] (d) ω 0 =0.1, Re[Z(iω 0 )] Fig. 12. Attractors of the generalized Duffing oscillator (5) under rectangular-pulse forcing, with decreasing Re[Z(iω 0 )] from (a) to (d). The angular frequency ω 0 of the rectangular-pulse forcing satisfies Re[Z(iω 0 )] < 0. Neighboring branches of the chaotic spiral attractor tightens and the spiral s armgrows gradually as Re[Z(iω 0 )] decreases. (a) Rectangular-pulse forcing j(t) =6s[sin(0.245t)]. Re[Z(iω 0 )] for ω 0 = Initial condition: x(0) = 0.5, y(0) = 0.5. (b) Rectangular-pulse forcing j(t) =6s[sin(0.2t)]. Re[Z(iω 0 )] for ω 0 =0.2. Initial condition: x(0) = 0.5, y(0) = 0.5. (c) Rectangular-pulse forcing j(t) =6s[sin(0.15t)]. Re[Z(iω 0 )] for ω 0 =0.15. Initial condition: x(0) = 0.5, y(0) = 0.5. (d) Rectangularpulse forcing j(t) =6s[sin(0.1t)]. Re[Z(iω 0 )] for ω 0 =0.1. Initial condition: x(0) = 0.5, y(0) =

17 Chaotic Oscillation via Edge of Chaos Criteria (a) j(t) =6sin(0.2t) (b) j(t) =6s[sin(0.2t)] Fig. 13. Average power P (t) of the generalized Duffing oscillator defined by Eq. (5). We set ω 0 = 0.2, which satisfies Re[Z(iω 0 )] < 0. (a) P (t) < 0fort 0 under sinusoidal forcing. The average power delivered from the forced signal is not dissipated in the chaotic oscillation. Here, the periodic forcing is given by j(t) = 6sin(0.2t). Initial condition: x(0) = 0.5, y(0) = 0.5. (b) P (t) > 0fort 0 under the rectangular-pulse. The average power delivered from the forced signal is dissipated in the chaotic oscillation. Here, the periodic forcing is given by j(t) = 6s[sin(0.2t)]. Initial condition: x(0) = 0.5, y(0) = 0.5. This figure shows that while the edge of chaos condition Re[Z(iω 0 )] < 0forsomeω 0 is a necessary condition in all of our simulations, for the average power P (t) of the oscillator to be eventually negative, it is not a sufficient condition. (a) j(t) =0 (b)j(t) =7sin(t) (c) j(t) =5s[sin(t)] Fig. 14. Trajectories of the generalized Duffing oscillator defined by Eq. (5). We set ω 0 = 1, which satisfies Re[Z(iω 0 )] > 0. (a) A trajectory of Eq. (5) tends to the asymptotically-stable equilibrium point (the origin) as t, if j(t) = 0. In this case, Eq. (5) is an autonomous system. Initial condition: x(0) = 0.5,y(0) = 0.5. (b) A chaotic trajectory of Eq. (5) driven by sinusoidal forcing j(t) = 7sin(t). Initial condition: x(0) = 0.5,y(0) = 0.5. (c) A chaotic trajectory of Eq. (5) driven by rectangular-pulse forcing j(t) = 5s[sin(t)]. Initial condition: x(0) = 0.5,y(0) =

18 M. Itoh & L. Chua (a) j(t) =0 (b)j(t) =7sin(t) (c) j(t) =5s[sin(t)] Fig. 15. Projection of the trajectories of Eq. (5) into the (ξ,η, ζ)-space via the coordinate transformation (80). We set ω 0 =1, which satisfies Re[Z(iω 0 )] > 0. Observe the difference among the chaotic trajectories in Figs. 9 and 15. Chaotic attractors in Figs. 15(b) and 15(c) look like twisted woolen yarns. The trajectory is coded using the rainbow color code for ζ, that is, the color evolves through violet, indigo, blue, green, yellow, orange and red, as ζ varies from its minimum to its maximum value. (a) A trajectory for j(t) = 0, which converges to a thick green circle with a radius of 5 after the transient regime. Since the trajectory on the (x, y)-plane tends to an asymptotically-stable equilibrium point (origin) as shown in Fig. 14(a), the corresponding trajectory in the (ξ,η,ζ)-space tends to a green circle. That is, the equilibrium point of Eq. (5) with j(t) = 0 is transformed into a green circle by the coordinate transformation (80) (see Appendix A for more details). Initial condition: x(0) = 0.5,y(0) = 0.5. The initial point (x(0),y(0)) = (0.5, 0.5) for Eq. (5) is projected into the point (ξ(0),η(0),ζ(0)) = (5.5, 0, 0.5) via the coordinate transformation (80). (b) A chaotic trajectory driven by sinusoidal forcing j(t) = 7 sin(t). Since Eq. (5) exhibits a chaotic oscillation on the (x, y)-plane as shown in Fig. 14(b), the corresponding trajectory in the (ξ,η,ζ)-space is also chaotic (see Appendix A for more details). Initial condition: x(0) = 0.5,y(0) = 0.5. (c) A chaotic trajectory driven by rectangular-pulse forcing j(t) =5s[sin(t)]. Since Eq. (5) exhibits a chaotic oscillation on the (x, y)-plane as shown in Fig. 14(c), the corresponding trajectory in the (ξ, η, ζ)-space is also chaotic (see Appendix A for more details). Initial condition: x(0) = 0.5,y(0) =

19 Chaotic Oscillation via Edge of Chaos Criteria in (a) and j(τ) =6s[sin(0.2τ)], (79) in (b), respectively. From our computer simulations, we obtain the following result: Average power P (t) forre[z(iω 0 )] < 0 Choose ω 0 =0.2 which satisfies Re[Z(iω 0 )] < 0. Then, the average power P (t) enteringthe one-port N satisfies: Sinusoidal forcing P (t) isnegative if time t is sufficiently large. That is, the average power is not dissipated in the chaotic circuit. In this case, the forced signal j(τ) =6sin(0.2τ) acts like a sink! Rectangular-pulse forcing P (t) ispositive if time t is sufficiently large. That is, the average power is dissipated in the chaotic circuit, and is delivered from the forced signal j(τ) = 6 s [ sin(0.2τ) ].Inthiscase,j(τ) acts like a source. Thus, we conclude as follows: Energy generation for Re[Z(iω 0 )] < 0 The generalized Duffing oscillator (5) is globally asymptotically stable without forcing. If a sinusoidal forcing is applied, then it can generate energy via chaotic oscillation. In this case, the oscillator is functioning as a source of energy, while the current source is functioning as a sink. Observe that the chaotic attractors in Figs. 15(b) and 15(c) look like twisted woolen yarns. We also show the Poincaré maps, and the average power in Figs. 16 and 17. Observe that the chaotic attractors are not tightly compressed. Furthermore, the average power P (t) entering the one-port N is positive if time t is large. That is, the average power which is delivered from the forced signal is dissipated in the chaotic circuit. (a) j(t) =7sin(t) Finally, let us study the case where the periodic forcing j(t) ineq.(5)satisfiesre[z(iω 0 )] > 0. Note that Eq. (5) can exhibit a chaotic oscillation, even if Re[Z(iω 0 )] > 0. We show the chaotic trajectories in Fig. 14. Here, we choose ω 0 = 1, which satisfies the above condition. We also show the trajectories projected into the (ξ,η,ζ)-space in Fig. 15, where the coordinate transformation is defined by ξ(t) =(x(t)+5)cos(t), η(t) =(x(t)+5)sin(t), ζ(t) =y(t). (80) (b) j(t) =5s[sin(t)] Fig. 16. Poincaré maps of the generalized Duffing oscillator defined by Eq. (5). We set ω 0 = 1, which satisfies Re[Z(iω 0 )] > 0. Observe that the chaotic attractors are not compressed tightly compared with those in Fig. 10. Thus, it slightly resembles the Poincaré section of the (nondissipative) Hénon Heiles Hamiltonian system. (a) Poincaré map driven by sinusoidal forcing j(t) = 7sin(t). Initial condition: x(0) = 0.5,y(0) = 0.5. (b) Poincaré map driven by rectangular-pulse forcing j(t) = 5s[sin(t)]. Initial condition: x(0) = 0.5,y(0) =

20 M. Itoh & L. Chua (a) j(t) =7sin(t) (b) j(t) =5s[sin(t)] Fig. 17. Average power P (t) of the generalized Duffing oscillator defined by Eq. (5). We set ω 0 = 1, which satisfies Re[Z(iω 0 )] > 0, and the periodic forcing signals are given by (a) j(t) =7sin(t) and(b)j(t) =5s[sin(t)]. Since P (t) > 0 for t 0, the average power delivered from the forced signal generator is dissipated in the chaotic circuit. In this case, the generalized Duffing oscillator circuit is acting as a sink. Initial condition for (a) and (b): x(0) = 0.5,y(0) = 0.5. We show the chaotic attractors in Figs. 18 and 19 for some other angular frequency ω 0,which satisfies Re[Z(iω 0 )] > 0. Observe that the chaotic attractor shrinks gradually until it degenerates into a periodic point, as Re[Z(iω 0 )] decreases. Furthermore, the chaotic attractor is not compressed under sinusoidal forcing. Similarly, the chaotic attractor does not spiral tightly under rectangular-pulse forcing. Thus, the chaotic oscillation for Re[Z(iω 0 )] < 0 becomes more complex than the oscillation for Re[Z(iω 0 )] > 0. We conclude as follows: Suppose that the angular frequency ω 0 of periodic forcing satisfies the edge of chaos condition (ii), that is, Re[Z(iω 0 )] < 0. Then, the chaotic attractor can be increasingly compressed until it looks like a one-dimensional curve, or it can spiral increasingly tightly and the spiral s arm grows gradually. (a) ω 0 =0.496, Re[Z(iω 0 )] 3.42 (b) ω 0 =0.51, Re[Z(iω 0 )] 2.27 Fig. 18. Attractors of the generalized Duffing oscillator (5) with decreasing Re[Z(iω 0 )] from (a) to (f). The sinusoidal forcing satisfies Re[Z(iω 0 )] > 0. Observe the difference among the chaotic attractors in Figs. 11 and 18. The chaotic attractor in Fig. 18 shrinks gradually until it degenerates into a periodic point with period-3 as Re[Z(iω 0 )] decreases. However, the chaotic attractor in Fig. 11 is compressed tightly until it looks like a one-dimensional curve as Re[Z(iω 0 )] decreases. (a) Sinusoidal forcing j(t) =5.5sin(0.496t). Re[Z(iω 0 )] 3.42 for ω 0 = Initial condition: x(0) = 0.5,y(0) = 0.5. (b) Sinusoidal forcing j(t) =5.5sin(0.51t). Re[Z(iω 0 )] 2.27 for ω 0 =0.51. Initial condition: x(0) = 0.5,y(0) = 0.5. (c) Sinusoidal forcing j(t) =5.5sin(0.524t). Re[Z(iω 0 )] 1.64 for ω 0 = Initial condition: x(0) = 0.5,y(0) = 0.5. (d) Sinusoidal forcing j(t) =5.5sin(0.5255t). Re[Z(iω 0 )] 1.58 for ω 0 = Initial condition: x(0) = 0.5,y(0) = 0.5. (f) Sinusoidal forcing j(t) =5.5sin( t). Re[Z(iω 0 )] 1.58 for ω 0 = Initial condition: x(0) = 0.5,y(0) = 0.5. (e) Sinusoidal forcing j(t) =5.5sin(0.53t). Re[Z(iω 0 )] 1.45 for ω 0 =0.53. Initial condition: x(0) = 0.5,y(0) =

21 Chaotic Oscillation via Edge of Chaos Criteria (c) ω 0 =0.524, Re[Z(iω 0 )] 1.64 (d) ω 0 =0.5255, Re[Z(iω 0 )] 1.58 (e) ω 0 = , Re[Z(iω 0 )] 1.58 (f) ω 0 =0.53, Re[Z(iω 0 )] 1.45 Fig. 18. (Continued)

22 M. Itoh & L. Chua (a) ω 0 =0.49, Re[Z(iω 0 )] 4.21 (b) ω 0 =0.51, Re[Z(iω 0 )] 2.27 (c) ω 0 =0.526, Re[Z(iω 0 )] 1.57 (d) ω 0 =0.5267, Re[Z(iω 0 )] 1.55 Fig. 19. Attractors of the generalized Duffing oscillator (5) under rectangular-pulse forcing, with decreasing Re[Z(iω 0 )] from (a) to (f), where Re[Z(iω 0 )] > 0. Observe the difference among the attractors in Figs. 12 and 19. The chaotic attractor in Fig. 19 shrinks gradually until it degenerates into a periodic point with period-4 as Re[Z(iω 0 )] decreases. However, the chaotic attractor in Fig. 12 does not shrink gradually, but the spiral s arm grows gradually as Re[Z(iω 0 )] decreases. (a) Rectangular-pulse forcing j(t) =5.5 s[sin(0.49t)]. Re[Z(iω 0 )] 4.21 for ω 0 =0.49. Initial condition: x(0) = 0.5, y(0) = 0.5. (b) Rectangular-pulse forcing j(t) =5.5 s[sin(0.51t)]. Re[Z(iω 0 )] 2.27 for ω 0 =0.51. Initial condition: x(0) = 0.5, y(0) = 0.5. (c) Rectangular-pulse forcing j(t) = 5.5 s[sin(0.526t)]. Re[Z(iω 0 )] 1.57 for ω 0 = Initial condition: x(0) = 0.5, y(0) = 0.5. (d) Rectangular-pulse forcing j(t) =5.5 s[sin(0.5267t)]. Re[Z(iω 0 )] 1.55 for ω 0 = Initial condition: x(0) = 0.5, y(0) = 0.5. (e) Rectangular-pulse forcing j(t) =5.5 s[sin( t)]. Re[Z(iω 0 )] 1.54 for ω 0 = Initial condition: x(0) = 0.5, y(0) = 0.5. (f) Rectangular-pulse forcing j(t) =5.5 s[sin(0.528t)]. Re[Z(iω 0 )] 1.51 for ω 0 = Initial condition: x(0) = 0.5, y(0) =

23 Chaotic Oscillation via Edge of Chaos Criteria (e) ω 0 = , Re[Z(iω 0 )] 1.54 (f) ω 0 =0.528, Re[Z(iω 0 )] Duffing Oscillator Consider the Duffing oscillator defined by ẋ = y, ẏ = 0.05y x 3 0.2x + j(t). Fig. 19. (81) Equation (81) can be realized by the circuit in Fig. 22. Its dynamics can be given by L 1 di 1 dt = v 1, C 1 dv 1 dt = i 1 G 1 v 1 i i 1 + j(t) where = G 1 v 1 i i 1 + j(t), (82) C 1 =1, L 1 =1, G 1 =0.05. (83) Substituting Eq. (83) into Eq. (81), we obtain di 1 dt = v 1, (84) dv 1 dt = 0.05v 1 i i 1 + j(t). Equations (81) and (84) are identical, if we rename the variables i 1 = x, v 1 = y. (85) Assume that j(t) = 0. Then, we obtain from Eq. (81) ẋ = y, ẏ = 0.05y x 3 0.2x. (86) (Continued) Equation (86) is an autonomous system, and it has only one equilibrium point at the origin. The Jacobian matrix J of the linearized system at the equilibrium point is given by [ ] 0 1 J =. (87) The characteristic equation associated with J is: Its eigenvalues are given by λ λ +0.2 =0. (88) λ = 0.05 ± i ± i (89) Thus, the equilibrium point of Eq. (86), namely, the origin, is locally asymptotically stable. We next show that the above equilibrium point is globally asymptotically stable. Let X(x, y) be the vector field of Eq. (86), that is, X(x, y) =(ẋ, ẏ) =(y, 0.05y x 3 0.2x). (90) Then, we obtain div X(x, y) = (91) Thus, Eq. (86) has no closed trajectories by Bendixson s criterion [Andronov et al., 1987]. Define the Lyapunov function by V (x, y) = 1 2 y x x4. (92)

24 M. Itoh & L. Chua Then, we obtain dv (x, y) dt = yẏ +0.2xẋ + x 3 ẋ = y( 0.05y x 3 0.2x)+0.2xy + x 3 y Fig. 20. Trajectory of the variational equation (95) with periodic forcing j(t) = 11.5cos(t). The trajectory tends to a periodic orbit (red), which starts from the initial condition: δx(0) = 0,δy(0) = 0. Fig. 21. Average power P (t) of the variational equation (95). P (t) is positive even if time t is sufficiently large. Here, the periodic forcing for Eq. (95) is given by j(t) =11.5cos(t). Initial condition: δx(0) = 0, δy(0) = 0. The current source in this case is the source of the energy dissipated inside the one-port N. = 0.05y 2 0. (93) Thus, V (x, y) decreases until the trajectory converges to the equilibrium point. It follows that the equilibrium point of Eq. (86) is globally asymptotically stable. Let δx and δy denote infinitesimal variables in the neighborhood of the point (x 0,y 0 )=(0, 0), namely, x(t) =x 0 + δx(t) =δx(t), y(t) =y 0 + δy(t) =δy(t). (94) From Eq. (81), we obtain the variational equation d(δx) dt = δy, d(δy) dt = 0.05δy 0.2δx + j(t). (95) Applying the Laplace transform to each term in Eq. (95), we obtain sˆx(s) δx(0) = ŷ(s), sŷ(s) δy(0) = 0.05ŷ(s) 0.2ˆx(s)+ĵ(s). (96) Assume that δx(0) = δy(0) = 0. Then we obtain from Eq. (96) ˆx(s) =ŷ(s) s, ŷ(s) = sĵ(s) s s (97) Fig. 22. Circuit model for Eq. (81). Parameters: C 1 =1,L 1 =1,G 1 =0.05. The nonlinear current-controlled current source is defined by i c = i i 1. The independent current source j(t) denotes a periodic forcing

25 The impedance Z(s) is defined by Z(s) = ŷ(s) ĵ(s) = It has the two poles: p 1 = i p 2 = 0.05 i s s s (98) i0.4465, i (99) They are in the open left-half plane. Thus, the Duffing oscillator satisfies the edge of chaos condition (i). Consider next the small-signal circuit model for Eq. (95). The impedance Z(s) defined by Eq. (98) can be recast into 1 Z(s) = s (100) 5s The impedance Z(s) of the linear circuit in Fig. 2 is given by Z(s) = L[v(t)] L[j s (t)] = 1, (101) 1 sc + G + sl + R where v(t) denotes the voltage across the capacitor C and j s (t) denotes the independent current source. Thus, Eq. (100) can be realized by the circuit in Fig. 2. Its parameters are given by C =1, L =5, G =0.05, R =0. (102) Chaotic Oscillation via Edge of Chaos Criteria Since R = 0, the linear circuit Fig. 2 can be simplified to the circuit shown in Fig. 23. Since its circuit parameters are all positive, this small-signal circuit model is passive. If we apply the periodic forcing j s (t) to the small-signal circuit model in Fig. 23, then the dynamics of this circuit is given by L di dt = v, that is, C dv dt = i Gv + j s(t), (103) di dt =0.2v, dv dt = i 0.05v + j s(t). (104) Applying the Laplace transform to each term in Eq. (104) under zero initial condition, we can obtain Eqs. (98) and (100). Here, we obtain Eq. (98) using the relationship among the state variables. If we set v =5 v and j s (t) =5j(t), then we obtain from Eq. (104) di dt = v, d v dt = 0.2i 0.05 v + j(t). (105) It is equivalent to Eq. (95). That is, we can obtain the relationship Hence, i(t) =δx(t), v(t) =δy(t). (106) L[v(t)] L[j s (t)] = L[5 v(t)] L[5j(t)] = L[ v(t)] L[j(t)] = L[δy(t)] L[j(t)] = ŷ(s) ĵ(s) = Z(s). (107) Fig. 23. Small-signal circuit model for Eq. (100) (left). This linearized circuit has an impedance defined by Z(s) = L[v(t)] L[j = s(t)] 1 sc + G + 1,whereC =1,L=5,G=0.05. The independent current source j s(t) denotes a periodic forcing. sl

26 M. Itoh & L. Chua Consider next Z(iω), which can be written as Z(iω) = iω 0.2 ω 2 + i0.05ω = 0.05ω2 + iω(0.2 ω 2 ) (0.2 ω 2 ) ω 2. (108) Its real part and imaginary part are given by Re[Z(iω)] = Im[Z(iω)] = 0.05ω 2 (0.2 ω 2 ) ω 2 0, ω(0.2 ω 2 ) (0.2 ω 2 ) ω 2. (109) Since Re[Z(iω)] 0, the edge of chaos condition (ii) cannot be applied to the Duffing oscillator (81). In other words, the linearized one-port connected across the current source j s (t) infig.23doesnot have an edge of chaos domain. This is the reason we added the term cx to the first equation of Eq. (81) in Sec. 2. We obtain from Eq. (97) ( ) s ŷ(s) = ĵ(s), ˆx(s) =ŷ(s) s s s +0.2 ( = 1 s s +0.2 ) ĵ(s). (110) In this case, if j(t) = A cos(ωt), then the trajectory tends to the periodic orbit defined by δy(τ) = y(τ) = A Z(iω) cos(ωτ + θ(ω)), δx(τ) =x(τ) =A Z 2 (iω) cos(ωτ + θ 2 (ω)), (111) as t, as shown in Fig. 20 for j(t) =11.5cos(t). Here, 0 τ 2π ω and Z(s) = Z 2 (s) = s s s +0.2, 1 s s +0.2, θ(ω) =argz(iω), θ 2(ω) =argz 2 (iω), ĵ(s) = L[j(t)] = L[A cos(ωt)] = As s 2 + ω 2. (112) Since the trajectory starting from the origin tends to this periodic orbit (red), the origin is no longer an equilibrium point of Eq. (95). 9 Let us study next the average power which is delivered from the periodic forcing current source. Since Z(s) is defined as an impedance, ĵ(s)andŷ(s) are the current and the voltage, respectively [see Eqs. (98), (106), and (107)]. Thus, we can define the instantaneous power entering the one-port N by p(t) = δy(t)j(t). (113) Let us choose ω = ω 0 > 0andobtain P n = 1 T = 1 T = 1 T nt (n 1)T nt (n 1)T nt (n 1)T p(τ)dτ δy(τ)j(τ)dτ A Z(iω) cos(ω 0 τ + θ(ω 0 ))A cos(ω 0 τ)dτ A2 2 Z(iω 0) cos θ(ω 0 ) = A2 2 Re[Z(iω 0)] > 0, (114) where θ(ω 0 ) = argz(iω 0 )andn is a sufficiently large integer. Thus, ( P (N) = 1 N ) P n > 0, (115) N n=1 for sufficiently large N. Since the continuous time average power is defined by P (t) = 1 t t 0 p(τ)dτ = 1 t t 0 δy(τ)j(τ)dτ, (116) it follows that P (t) > 0, as confirmed in Fig. 21. Here, the periodic forcing of Eq. (97) is given by j(t) = 11.5cos(t). Thus, the variational equation (95) dissipates the periodic forcing energy. Let us study next the behavior of Eq. (81) of the circuit shown in Fig. 22. We first show the trajectory of Eq. (81) with j(t) = 0 in Fig. 24(a). In this case, Eq. (81) is autonomous. All trajectories tend to a globally asymptotically-stable equilibrium point (origin) as t. We next show the trajectories of Eq. (81) with the periodic forcing j(t) 0 9 The concept of an equilibrium point is valid only for autonomous systems as stated in Sec

27 Chaotic Oscillation via Edge of Chaos Criteria (a) j(t) =0 (b)j(t) =11.5cos(t) (c) 16 s[cos(t)] Fig. 24. Trajectories of the Duffing oscillator defined by Eq. (81), where Re[Z(iω)] 0 for all ω. (a) A trajectory of Eq. (81) tends to the asymptotically-stable equilibrium point (the origin) as t,ifj(t) = 0. In this case, Eq. (81) is an autonomous system. Initial condition: x(0) = 1,y(0) = 1. (b) A chaotic trajectory of Eq. (81) under the sinusoidal forcing j(t) =11.5cos(t). Initial condition: x(0) = 1,y(0) = 1. (c) A chaotic trajectory of Eq. (81) under the rectangular-pulse forcing j(t) = 16 s[cos(t)]. Initial condition: x(0) = 1,y(0) = 1. in Figs. 24(b) and 24(c). In this case, Eq. (81) is nonautonomous. The periodic forcing signals are given, respectively, by and j(t) =11.5cos(t), (117) j(t) =16s[cos(t)]. (118) Here, we choose ω = 1, since no angular frequency ω exists, which satisfies the edge of chaos condition (ii). Observe that Eq. (81) exhibits a chaotic oscillation when driven by a periodic forcing, as shown in Figs. 24(b) and 24(c). We next show the trajectories projected into the (ξ,η,ζ)-space in Fig. 25, where the coordinate transformation is given by ξ(t) =(x(t)+5)cos(t), η(t) =(x(t)+5)sin(t), ζ(t) =y(t). (119) We also show their Poincaré maps in Fig. 26. Our computer simulations show that Eq. (81) can exhibit chaotic oscillation even if Re[Z(iω)] 0 for all ω. We also found that if Re[Z(iω 0 )] is decreased slightly, the chaotic attractor disappears, and a periodic oscillation appears instead, because the chaotic regime of the Duffing oscillator (81) is very small. From our computer simulations, we obtain

28 M. Itoh & L. Chua (a) j(t) =0 (b)j(t) =11.5cos(t) (c) 16 s[cos(t)] Fig. 25. Projection of the trajectories of Eq. (81) into the (ξ,η, ζ)-space via the coordinate transformation (119). Observe the difference among the three orbits. Recall that Re[Z(iω)] 0 for all ω. Chaotic attractors in Figs. 25(b) and 25(c) look like loosely twisted woolen yarns. The trajectory is coded using the rainbow color code for ζ, that is, the color evolves through violet, indigo, blue, green, yellow, orange and red, as ζ varies from its minimum to its maximum value. (a) A trajectory of Eq. (81) with j(t) = 0, which converges to a thick green circle with a radius of 5 after the transient regime tends to zero. Since the trajectory on the (x, y)-plane tends to an asymptotically-stable equilibrium point (origin) as shown in Fig. 24(a), the corresponding trajectory in the (ξ, η, ζ)-space tends to a green circle. That is, the equilibrium point is transformed into a green circle by the coordinate transformation (119) (see Appendix A for more details). Initial condition: x(0) = 1,y(0) = 1. The initial point (x(0),y(0)) = (1, 1) for Eq. (81) is projected into the point (ξ(0),η(0),ζ(0)) = (6, 0, 1) via the coordinate transformation (80). (b) A chaotic trajectory of Eq. (81) under the sinusoidal forcing j(t) =11.5cos(t). Since Eq. (81) exhibits a chaotic oscillation on the (x, y)-plane as shown in Fig. 24(b), the corresponding trajectory in the (ξ,η,ζ)-space is also chaotic (see Appendix A for more details). Initial condition: x(0) = 1,y(0) = 1. (c) A chaotic trajectory of Eq. (81) under the rectangular-pulse forcing j(t) = 16s[cos(t)]. Since Eq. (81) exhibits a chaotic oscillation on the (x, y)-plane as shown in Fig. 24(c), the corresponding trajectory in the (ξ, η, ζ)-space is also chaotic (see Appendix A for more details). Initial condition: x(0) = 1,y(0) =

29 Chaotic Oscillation via Edge of Chaos Criteria (a) j(t) =11.5cos(t) (b) 16s[cos(t)] Fig. 26. Poincaré maps of the Duffing oscillator defined by Eq. (81), where Re[Z(iω)] 0 for all ω. (a)poincaré map under the sinusoidal forcing j(t) = 11.5cos(t). Initial condition: x(0) = 1,y(0) = 1. (b) Poincaré map under the rectangular-pulse forcing j(t) = 16s[cos(t)]. Initial condition: x(0) = 1,y(0) = 1. the following result: Oscillation for Duffing oscillator (81) For small periodic forcing, the Duffing oscillator (81) exhibits a small periodic orbit (periodic response). If we increase the amplitude of the periodic forcing, the periodic response grows, and can evolve into a chaotic trajectory. Observe that y(t) and j(t) can be identified as the voltage v 1 (t) across the capacitor C 1 and the current j(t) of the independent current source, respectively [see Fig. 22 and Eqs. (82) and (85)]. That is, y(t) andj(t) are equivalent to the terminal voltage and current of the independent current source in Fig. 22, respectively. Thus, we can define the instantaneous power entering the one-port N at time τ by y(τ)j(τ), and the average power P (t) by P (t) = 1 t y(τ)j(τ)dτ. (120) t 0 We show the average power P (t) of Eq. (81) in Fig. 27. Here, the periodic forcing signal in Eq. (5) is given respectively by j(t) = 11.5cos(t) and (a) j(t) =11.5cos(t) (b) 16s[cos(t)] Fig. 27. Average power P (t) of the Duffing oscillator defined by Eq. (81), where Re[Z(iω)] 0 for all ω. (a) P (t) > 0for t>0under the sinusoidal forcing. The average power delivered from the forced signal is dissipated in the chaotic circuit. Here, the periodic forcing is given by j(t) =11.5cos(t). Initial condition: x(0) = 1,y(0) = 1. (b) P (t) < 0fort 0 under the rectangular-pulse forcing. The average power in this case is generated by the chaotic one-port N, and the current source acts like a sink. Here, the periodic forcing is given by 16 s[cos(t)]. Initial condition: x(0) = 1,y(0) =

30 M. Itoh & L. Chua 16 s[cos(t)]. Observe that the average power P (t) under sinusoidal forcing is positive, even if time t is sufficiently large. However, in the case of a periodic rectangular-pulse forcing, the average power P (t) entering the one-port N becomes negative. Average power of the Duffing oscillator (81) Sinusoidal forcing The average power P (t) ispositive. That is, the average power is dissipated in the chaotic circuit, which is delivered from the forced sinusoidal signal j(t) =11.5cos(t). Rectangular-pulse forcing The average power P (t) isnegative if time t is sufficiently large. That is, the average power is not dissipated in the chaotic circuit. The forced periodic rectangularpulse signal 16 s[cos(t)] in this case acts like a sink. The source of the energy in this case comes from the nonlinear currentcontrolled current source defined in Fig FitzHugh Nagumo Model The FitzHugh Nagumo model is a two-dimensional simplification of the Hodgkin Huxley model. The dynamics is given by ẋ = x x3 3 y + I, ẏ =0.08(x y), (121) where I is the external stimulus. Observe that Eq. (121) is extremely similar to Eq. (10). Thus, the Duffing oscillator is closely related to the FitzHugh Nagumo model. Let us study the equilibrium point of Eq. (121), which satisfies x x3 3 y + I =0, x y =0. (122) From the second equation in Eq. (122), we obtain y = x (123) 0.8 Substituting this equation into the first equation in Eq. (122), we obtain x x3 3 x I =0. (124) 0.8 It can be recast into the form I = f(x) = x x (125) Since f(x) is a strictly monotone-increasing func- Eq. (124) has only one solution for any given tion, constant I. Let (x 0,y 0 ) be the equilibrium point of Eq. (121) with I = f(x 0 ). The Jacobian matrix J at this point is given by J = [ 1 x ]. (126) The characteristic equation of J can be written as λ 2 +(x )λ x = 0, (127) and its eigenvalues are given by where α = x , λ = α ± D, (128) 2 D =(x ) 2 4(0.064x ) = x x (129) If α>0 and D<0, then the eigenvalues have negative real part, and the equilibrium point is asymptotically stable. This condition is easily satisfied, for example, if we choose x 0 = In this case, we obtain I = f(0.977) Consider next the FitzHugh Nagumo model with periodic forcing [Itoh & Chua, 2007] ẋ = x x3 3 y + I + j(t), ẏ =0.08(x y), (130) where I denotes a constant current source and j(t) denotes a periodic forcing. Equation (130) can be realized by the circuit in Fig. 28. Its dynamics is given by dv 1 C 1 dt = v 1 v i 1 + I + j(t), L 1 di 1 dt = v 1 + E R 1 i 1, (131)

31 Chaotic Oscillation via Edge of Chaos Criteria Fig. 28. Circuit model for Eq. (130). Parameters: C 1 =1,L 1 = ,R 1 =0.8,E =0.7, I is a constant. The nonlinear resistor is defined by i R = v R 3 3 v R. The direct current (DC) source is denoted by I. The battery is denoted by E. The independent current source j(t) denotes a periodic forcing. where C 1 =1, L 1 = , R 1 =0.8, E=0.7. (132) Substituting Eq. (132) into Eq. (131), we obtain 3 dv 1 dt = v 1 v 1 3 i 1 + I + j(t), di 1 dt =0.08(v i 1 ). (133) Equations (130) and (133) are identical, if we rename the variables v 1 = x, i 1 = y. (134) Let δx and δy denote infinitesimal variables in the neighborhood of the equilibrium point (x 0,y 0 ) of Eq. (121), namely, x(t) =x 0 + δx(t), y(t) =y 0 + δy(t). (135) From Eq. (130), we obtain the variational equation d(δx) =(1 x 2 0 )δx δy + j(t), dt (136) d(δy) =0.08(δx 0.8δy). dt Let us define the Laplace transform of δx, δy, and j(t) by ˆx(s) = 0 δx(t)e st dt, ŷ(s) = ĵ(s) = 0 0 j(t)e st dt, δy(t)e st dt, (137) respectively. Applying the Laplace transform to each term in Eq. (136), we obtain sˆx(s) =(1 x 0 2 )ˆx(s) ŷ(s)+ĵ(s), sŷ(s) =0.08(ˆx(s) 0.8ŷ(s)). (138) Here, we assumed zero initial conditions, that is, δx(0) = 0 and δy(0) = 0. From Eq. (135), we obtain x(0) = x 0, y(0) = y 0. (139) Thus, the initial point is the equilibrium point (x 0,y 0 ). Solving the second equation in Eq. (138) for ŷ(s), we obtain ŷ(s) = 0.08ˆx(s) s (140) Substituting this equation into the first equation in Eq. (138), we obtain ( s (1 x 2 0 ) ) ˆx(s) =ĵ(s), (141) s that is, (s + x 2 0 1)(s ) ˆx(s) =ĵ(s). s (142) It can be written as s 2 +(x )s x ˆx(s)=ĵ(s). s (143)

32 M. Itoh & L. Chua Consider next the impedance Z(s) defined by Z(s) = ˆx(s) ĵ(s) = s s 2 +(x )s x (144) In this case, j(t) andx(t) are the current and the voltage of the one-port N connected across j(t) in Fig. 28. The impedance Z(s) has two poles: where α = x , p 1 = α + D, 2 p 2 = α D, 2 D =(x ) 2 4(0.064x ) = x x (145) (146) As stated before, if α > 0andD < 0, they are in the open left-half plane. For example, we can choose x 0 =0.977, that is, α = > 0and D < 0. If we assume that the periodic forcing is given Aω by j(t) = A sin(ωt), then ĵ(s) = s 2 + ω 2. Its response ˆx(s) andŷ(s) isgivenby ˆx(s) =Z(s)j(s) and Aω = Z(s) s 2 + ω 2 ( ) s = s 2 +(x )s x ( ) Aω s 2 + ω 2, (147) ŷ(s) = 0.08ˆx(s) s ( ) ( ) 0.08 Aω = Z(s) s s 2 + ω 2 ( ) 0.08 = s 2 +(x )s x ( ) Aω s 2 + ω 2, (148) respectively. As stated in Secs. 2.6 and 2.7, if we define θ(ω) by θ(ω) =argz(iω), (149) and if α>0andd<0, we obtain δx(t) =x(t) x 0 A Z(iω) sin(ωt + θ(ω)), (150) as t, since the two poles p 1 and p 2 have a negative real part. Similarly, from Eqs. (144) and (148), we obtain δy(t) =y(t) y 0 A Z 2 (iω) sin(ωt + θ 2 (ω)), (151) where Z 2 (iω) 0.08 = s 2 +(x )s x , s=iω θ 2 (ω) =argz 2 (iω). Z(iω) = and LetusnextstudytherealpartofZ(iω): = Re[Z(iω)] (152) iω ω 2 + i(x )ω x iω (0.064x ω 2 )+i(x )ω, (153) = 0.064(0.064x ω 2 )+ω 2 (x ) (0.064x ω 2 ) 2 +(x ) 2 ω 2 = 0.064(0.064x ) + ω 2 (x 0 2 1) (0.064x ω 2 ) 2 +(x ) 2 ω 2. (154) In order for Re[Z(iω)] to be negative, x 0 must satisfy x 0 2 < 1.Ifwechoosex 0 = and ω =0.5, then Re[Z(iω)] < 0 (see Fig. 30). Thus, the following edge of chaos criteria [Chua, 2005, 2013] is

33 Chaotic Oscillation via Edge of Chaos Criteria satisfied: Condition (i) All poles of Z(s) are on the open lefthalf plane. Condition (ii) Re[Z(iω)] < 0 for at least one frequency ω = ω 0. Since j(t) andx(t) are the current and the voltage of the one-port N [see Eq. (134)], we can define the small-signal instantaneous power entering the one-port N by p(t) = δx(t)j(t) =(x(t) x 0 )j(t). (155) If we choose ω = ω 0, which satisfies Re[Z(iω 0 )] < 0, then we obtain P n = 1 T 1 T nt (n 1)T T 0 p(τ)dτ = 1 T nt (n 1)T {A Z(iω 0 ) sin(ω 0 τ + θ(ω 0 ))} {A sin(ω 0 τ)}dτ δx(τ)j(τ)dτ = A2 2 Z(iω 0) cos θ(ω 0 )= A2 2 Re[Z(iω 0)] < 0, (156) where n is a sufficiently large integer and T = 2π. ω 0 That is, the average power entering the one-port N satisfies P (N) = 1 N P n < 0, (157) N n=1 for sufficiently large N. Consider next the small-signal circuit model for Eq. (136). The impedance Z(s) defined by Eq. (98) can be recast into 1 Z(s) = s (1 x 2 0 ) s =. (158) s +(x ) s +0.8 Fig. 29. Small-signal circuit model for Eq. (158) (left). This circuit has the impedance defined by Z(s) = L[v(t)] = L[j s(t)] 1,whereC =1,L =12.5,G = x < sc + G + sl + R 0,R =0.8. The independent current source j s(t) denotes a periodic forcing. Thus, the parameters of the small-signal circuit model in Fig. 2 is given by C =1, L =12.5, G = x 2 0 1, R =0.8. (159) That is, the small-signal circuit model is given by the circuit in Fig. 29. Observe that this circuit has a locally-active element G with a negative conductance equal to x 0 2 1[S], where S denotes the unit in Siemens, because x 0 must satisfy x 0 2 < 1 as stated before. If we apply the periodic forcing j s (t) to the small-signal circuit model in Fig. 29, then the dynamics of this circuit is given by C dv dt = i Gv + j s(t), L di dt = v Ri. (160) Substituting Eq. (159) into Eq. (160), we obtain dv dt = i (x 0 2 1)v + j s (t), di =0.08(v +0.8i), dt (161) which is equivalent to Eq. (136). That is, we obtain the relation i(t) =δy(t), v(t) =δx(t), j s (t) =j(t). (162)

34 M. Itoh & L. Chua Fig. 30. Real part of the small-signal impedance Z(iω) frequency response for x 0 = Hence, L[v(t)] L[j s (t)] = L[v(t)] L[j(t)] = L[δx(t)] L[j(t)] = ˆx(s) ĵ(s) = Z(s), (163) where L denotes the Laplace transform Computer simulations Case 1. I =1.43 We first show the trajectory of Eq. (130) with j(t) =0andI =1.43 in Fig. 31(a). In this case, Eq. (130) is autonomous. The trajectory tends to an asymptotically-stable equilibrium point (x 0,y 0 ) ( , ) as t. We next show the trajectories of Eq. (130) with the periodic forcing j(t) 0 in Figs. 31(b) and 31(c), where j(t) is given by the sinusoidal function j(t) = sin(0.5t), (164) and the rectangular-pulse function j(t) = s[sin(0.5t)]. (165) Here, we set ω 0 = 0.5, which satisfies the condition (ii) of the edge of chaos criteria, that is, Re[Z(iω 0 )] < 0. In order to view their trajectories from a different perspective, let us project the trajectories into the (ξ,η,ζ)-space via the coordinate transformation: ξ(t) =(x(t) + 10) cos(0.5t), η(t) =(x(t) + 10) sin(0.5t), ζ(t) =y(t). (166) We show the projected trajectories in Fig. 32. Observe the difference among the three orbits. We also show their Poincaré maps in Fig. 33. Note that we have to find the amplitude A of the periodic forcing experimentally, in which Eq. (130) exhibits chaotic behavior. Furthermore, by adjusting the value of A, we can observe the coexistence of two attractors, that is, a small periodic orbit (periodic response) in the neighborhood of (x 0,y 0 ) and a chaotic attractor, as shown in Fig. 34. Here, (x 0,y 0 ) ( , ) denotes the equilibrium point for j(t) = 0. Recall x(t) andj(t) are the voltage v 1 (t) across the capacitor C 1 and the current j(t) of the independent current source, respectively [see Fig. 28 and Eqs. (131) and (134)]. That is, x(t) and j(t) are equivalent to the terminal voltage and current of the independent current source in Fig. 28, respectively. Thus, we can define the instantaneous power entering the one-port N at time τ by x(τ)j(τ), and the average power P (t) by P (t) = 1 t t 0 x(τ) j(τ) dτ. (167) Observe that in view of the reference current direction and voltage polarity shown in Fig. 28, P (t) > 0 implies energy is dissipated in the one-port circuit at time t. Conversely, P (t) < 0 implies energy is dissipated in the current source. We show the average power P (t) for Eq. (130) in Fig. 35. The periodic forcing is given by j(t) = sin(0.5t). In this case, the average power P (t) becomes negative if time t is sufficiently large. That is, the average power from the forced signal is not dissipated in the chaotic circuit. However, in the case of the rectangular-pulse j(t) = 0.04 s[sin(0.5t)], the average power P (t) becomes positive, even if

35 Chaotic Oscillation via Edge of Chaos Criteria (a) j(t) = 0 (b)j(t) = sin(0.5t) (c) j(t) = s[sin(0.5t)] Fig. 31. Trajectories of the forced FitzHugh Nagumo model defined by Eq. (130) with I =1.43. We set ω 0 =0.5, which satisfies Re[Z(iω 0 )] < 0. (a) A trajectory of Eq. (130) tends to the asymptotically-stable equilibrium point (marked blue) as t if j(t) = 0. The coordinate of the equilibrium point is described by (x 0,y 0 ) ( , ). In this case, Eq. (130) is an autonomous system. Initial condition: x(0) = 0.5,y(0) = 0.5. (b) A chaotic trajectory of Eq. (130) under the sinusoidal forcing j(t) = sin(0.5t). Initial condition: x(0) = 0.5,y(0) = 0.5. (c) A chaotic trajectory of Eq. (130) under the rectangular-pulse forcing j(t) = s[sin(0.5t)]. Initial condition: x(0) = 0.5,y(0) =

36 M. Itoh & L. Chua (a) j(t) = 0 (b)j(t) = sin(0.5t) (c) j(t) = s[sin(0.5t)] Fig. 32. Projection of the trajectories of Eq. (130) into the (ξ,η,ζ)-space via the coordinate transformation (166). Observe the difference among three orbits. We set ω 0 =0.5, which satisfies Re[Z(iω 0 )] < 0. The trajectory is coded using the rainbow color code for ζ, that is, the color evolves through violet, indigo, blue, green, yellow, orange and red, as ζ varies from its minimum to its maximum value. (a) A trajectory for j(t) = 0, which converges to a thick red circle after the transient regime. Since the trajectory on the (x, y)-plane tends to an asymptotically-stable equilibrium point, the corresponding trajectory in the (ξ,η,ζ)-space tends to a red circle (see Appendix A for more details). Initial condition: x(0) = 0.5,y(0) = 0.5. The initial point (x(0),y(0)) = (0.5, 0.5) for Eq. (130) is projected into the point (ξ(0),η(0),ζ(0)) = (10.5, 0, 0.5) via the coordinate transformation (166). (b) A chaotic trajectory under the sinusoidal forcing j(t) = sin(0.5t). Since Eq. (130) exhibits a chaotic oscillation on the (x, y)-plane, the corresponding trajectory in the (ξ,η,ζ)-space is also chaotic (see Appendix A for more details). We deleted the transient trajectory to view the chaotic attractor clearly. The red portion of trajectory is not a periodic orbit, but a part of chaotic orbit. Initial condition: x(0) = 0.5,y(0) = 0.5. (c) A chaotic trajectory under the rectangular-pulse forcing j(t) = s[sin(0.5t)]. We can observe a gap which is not filled with trajectories. Since Eq. (130) exhibits a chaotic oscillation on the (x, y)-plane, the corresponding trajectory in the (ξ,η,ζ)-space is also chaotic (see Appendix A for more details). We deleted the transient trajectory to view the chaotic attractor clearly. The red portion of trajectory is not a periodic orbit, but a part of chaotic orbit. Initial condition: x(0) = 0.5,y(0) =

37 Chaotic Oscillation via Edge of Chaos Criteria (a) j(t) = sin(0.5t) (b) j(t) = s[sin(0.5t)] Fig. 33. Poincaré maps of the forced FitzHugh Nagumo model defined by Eq. (130) with I = We set ω 0 = 0.5, which satisfies Re[Z(iω 0 )] < 0. Many folding loci can be observed in the partial enlarged view of the Poincaré maps.(a) Poincaré map under the sinusoidal forcing j(t) = sin(0.5t) (left), and its partial enlarged view (right). Initial condition: x(0) = 0.5,y(0) = 0.5. (b) Poincaré map under the rectangular-pulse forcing j(t) = s[sin(0.5t)], and its partial enlarged view (right). Initial condition: x(0) = 0.5,y(0) =

38 M. Itoh & L. Chua Fig. 34. Coexistence of two attractors in the forced FitzHugh Nagumo model defined by Eq. (130) with I = A chaotic attractor (red) and a small periodic orbit (purple) coexist. The right figure is a partially enlarged view. The sinusoidal forcing is given by j(t) = sin(0.5t), where ω 0 =0.5 and it satisfies Re[Z(iω 0 )] < 0. Initial condition for small periodic orbit: x(0) = 0.95,y(0) = Initial condition for chaotic attractor: x(0) = 0.5,y(0) = 0.5. (a) j(t) = sin(0.5t) (b) j(t) = s[sin(0.5t)] Fig. 35. Average power P (t) of the forced FitzHugh Nagumo model defined by Eq. (130) with I =1.43. We set ω 0 =0.5, which satisfies Re[Z(iω 0 )] < 0. (a) P (t) < 0fort 0 under sinusoidal forcing. The average power from the forced signal is not dissipated in the chaotic circuit. Here, the periodic forcing given by j(t) = sin(0.5t) acts as s sink. Initial condition: x(0) = 0.5,y(0) = 0.5. (b) P (t) > 0fort>0 under the rectangular-pulse forcing. The average power delivered from the forced signal is dissipated in the one-port N. Here, the periodic forcing is given by j(t) = s[sin(0.5t)]. Initial condition: x(0) = 0.5,y(0) =

39 Chaotic Oscillation via Edge of Chaos Criteria Re[Z(iω 0 )] < 0. Thus, we conclude as follows: Average power P (t) fori =1.43 Choose ω 0 =0.5 which satisfies Re[Z(iω 0 )] < 0. Then, the average power P (t) satisfies: sinusoidal forcing The forced FitzHugh Nagumo model (130) does not dissipate average power P (t) from the forced signal j(t) = sin(0.5t). Thus, it can generate energy by chaotic oscillation. rectangular-pulse forcing The forced FitzHugh Nagumo model (130) dissipates a positive average power P (t), which is delivered from the current source j(t) = s[sin(0.5t)]. Finally, let us study the case where Re[Z(iω 0 )] > Let us choose ω 0 = 0.2, which satisfies Re[Z(iω 0 )] > 0. We show the trajectories of Eq. (130) with the periodic forcing j(t) in Fig. 36. In order to view their trajectories from a different perspective, we project the trajectories into the (ξ,η,ζ)-space via the coordinate transformation: ξ(t) =(x(t) + 10) cos(0.2t), (168) η(t) =(x(t) + 10) sin(0.2t), ζ(t) = y(t). We show the projected trajectories in Fig. 37. Compare the orbits in Fig. 37 with those in Fig. 32. In Figs. 37(b) and 37(c), we can observe a gap, which is not filled with trajectories. We also show the Poincaré maps in Fig. 38. Compare the attractors in Fig. 38 with those in Fig. 33. Observe that if Re[Z(iω 0 )] < 0, the behavior of the chaotic attractors becomes more complex, that is, they exhibit a continuum of folding loci. Case 2. I = Let us choose ω 0 = 0.5 andi = , which satisfies the edge of chaos criteria (see Fig. 39). If j(t) = 0, Eq. (130) has an asymptotically-stable equilibrium point (blue), a stable limit cycle (red), and an unstable limit cycle (green) as shown in Fig. 40(a). 11 In this case, the autonomous circuit (with j(t) = 0) exhibits a subcritical Hopf bifurcation [Guckenheimer & Holmes, 1983]. If we apply theperiodicforcingj(t), Eq. (5) exhibits a chaotic behavior, as shown in Figs. 40(b) and 40(c). Here, j(t) is given, respectively, by the sinusoidal function j(t) = sin(0.5t), (169) and the rectangular-pulse function j(t) = 0.04 s[sin(0.5t)]. (170) In order to view their trajectories from a different perspective, let us project the trajectories into the (ξ,η,ζ)-space via the coordinate transformation: ξ(t) =(x(t) + 10) cos(0.5t), (171) η(t) =(x(t) + 10) sin(0.5t), ζ(t) = y(t). We show the projected trajectories in Fig. 41. Compare the orbits in Fig. 41 with those in Fig. 32. We also show their Poincaré maps in Fig. 42. By adjusting the value of A, we can observe the coexistence of two attractors, that is, a small periodic orbit (periodic response) in the neighborhood of (x 0,y 0 ) and a chaotic attractor, as shown in Fig. 43. Here, (x 0,y 0 ) (0.97, 2.087) denotes the equilibrium point for j(t) = 0. We next show the average power P (t) for Eq. (130) in Fig. 44. It becomes negative if time t is sufficiently large, where the periodic forcing is given by j(t) =0.025 sin(0.5t). However, in the case of the rectangular-pulse forcing j(t) = 0.04 s[sin(0.5t)], the average power P (t) entering the one-port N is positive, even if Re[Z(iω 0 )] < 0. Average power P (t) fori = Choose ω 0 =0.5 which satisfies Re[Z(iω 0 )] < 0. Then, the average power P (t) satisfies: sinusoidal forcing The one-port N described by the forced FitzHugh Nagumo model (130) does not dissipate average power P (t) from the forced signal j(t) = sin(0.5t). On the contrary, it generates energy by chaotic oscillations via a locally-active circuit element inside the one-port N. rectangular-pulse forcing The one-port N described by the forced FitzHugh Nagumo model (130) dissipates average power P (t) delivered from the forced signal j(t) = 0.04 s[sin(0.5t)]. 10 Note that Eq. (130) can exhibit a chaotic oscillation, even if Re[Z(iω 0 )] > Limit cycle is defined only for autonomous systems. In this case, Eq. (130) is the autonomous system, since j(t) =

40 M. Itoh & L. Chua (a) j(t) = 0 (c) j(t) = 0.04 s[sin(0.2t)] (b)j(t) = sin(0.2t) Fig. 36. Trajectories of the forced FitzHugh Nagumo model defined by Eq. (130) with I =1.43. We set ω 0 =0.2, which satisfies Re[Z(iω 0 )] > 0. (a) A trajectory of Eq. (130) tends to the asymptotically-stable equilibrium point (marked blue) as t if j(t) = 0. The coordinate of the equilibrium point is given by (x 0,y 0 ) ( , ). In this case, Eq. (130) is an autonomous system. Initial condition: x(0) = 0.5,y(0) = 0.5. (b) A chaotic trajectory of Eq. (130) under the sinusoidal forcing j(t) = sin(0.2t). Initial condition: x(0) = 0.5,y(0) = 0.5. (c) A chaotic trajectory of Eq. (130) under the rectangular-pulse forcing j(t) = 0.04 s[sin(0.2t)]. Initial condition: x(0) = 0.5,y(0) =

41 Chaotic Oscillation via Edge of Chaos Criteria (a) j(t) = 0 (b)j(t) = sin(0.2t) (c) j(t) = 0.04 s[sin(0.2t)] Fig. 37. Trajectories of Eq. (130), which are projected into the (ξ,η,ζ)-space via the coordinate transformation (168). We set ω 0 =0.2, which satisfies Re[Z(iω 0 )] > 0. Observe the difference among the trajectories in Figs. 32 and 37. We can observe a wide gap in Figs. 37(b) and 37(c), which is not filled with trajectories. The trajectory is coded using the rainbow color code for ζ, that is, the color evolves through violet, indigo, blue, green, yellow, orange and red, as ζ varies from its minimum to its maximum value. (a) A trajectory for j(t) = 0, which converges to a thick red circle after the transient regime. Since the trajectory on the (x, y)-plane tends to an asymptotically-stable equilibrium point as shown in Fig. 36(a), the corresponding trajectory in the (ξ,η, ζ)-space tends to a red circle (see Appendix A for more details). Initial condition: x(0) = 0.5,y(0) = 0.5. The initial point (x(0),y(0)) = (0.5, 0.5) for Eq. (130) is projected into the point (ξ(0),η(0),ζ(0)) = (10.5, 0, 0.5) via the coordinate transformation (168). (b) A chaotic trajectory under the sinusoidal forcing j(t) = sin(0.2t). Since Eq. (130) exhibits a chaotic oscillation on the (x, y)-plane as shown in Fig. 36(b), the corresponding trajectory in the (ξ,η,ζ)-space is also chaotic (see Appendix A for more details). We deleted the transient trajectory to view the chaotic attractor clearly. The red portion of trajectory is not a periodic orbit, but a part of chaotic orbit. Initial condition: x(0) = 0.5,y(0) = 0.5. (c) A chaotic trajectory under the rectangular-pulse forcing j(t) = 0.04 s[sin(0.2t)]. Since Eq. (130) exhibits a chaotic oscillation on the (x, y)-plane as shown in Fig. 36(c), the corresponding trajectory in the (ξ,η,ζ)-space is also chaotic (see Appendix A for more details). We deleted the transient trajectory to view the chaotic attractor clearly. The red portion of trajectory is not a periodic orbit, but a part of chaotic orbit. Initial condition: x(0) = 0.5,y(0) =

42 M. Itoh & L. Chua (a) j(t) = sin(0.2t) (b) j(t) = 0.04 s[sin(0.2t)] Fig. 38. Poincaré maps of Eq. (130) with I =1.43. We set ω 0 =0.2, which satisfies Re[Z(iω 0 )] > 0. Compare the partially enlarged view in Fig. 38 with that in Fig. 33. Observe that the folds in Fig. 38 are more visible than those in Fig. 33. (a) Poincaré map under the sinusoidal forcing j(t) = sin(0.2t) (left) and its partial enlarged view (right). Initial condition: x(0) = 0.5,y(0) = 0.5. (b) Poincaré map under the rectangular-pulse forcing j(t) = 0.04 s[sin(0.2t)] (left) and its partial enlarged view (right). Initial condition: x(0) = 0.5,y(0) = 0.5. Fig. 39. Small-signal impedance Z(iω) frequency response for I = In this case, x

43 Chaotic Oscillation via Edge of Chaos Criteria (a) j(t) = 0 (c) j(t) = 0.04 s[sin(0.5t)] (b)j(t) = sin(0.5t) Fig. 40. Trajectories of the forced FitzHugh Nagumo model defined by Eq. (130) with I = We set ω 0 =0.5, which satisfies Re[Z(iω 0 )] < 0. (a) Equation (130) with j(t) = 0 has an asymptotically-stable equilibrium point (blue), a stable limit cycle (red), and an unstable limit cycle (green) as its limit set. The coordinate of the asymptotically-stable equilibrium point (blue) is given by (x 0,y 0 ) (0.97, 2.087). In this case, Eq. (130) is an autonomous system. We calculate the unstable limit cycle with reverse time scaling. Initial condition for a stable limit cycle: x(0) = 0.5,y(0) = 0.5. Initial condition for an unstable limit cycle: x(0) = 0.97,y(0) = (b) A chaotic trajectory of Eq. (130) under the sinusoidal forcing j(t) = sin(0.5t). Initial condition: x(0) = 0.5,y(0) = 0.5. (c) A chaotic trajectory of Eq. (130) under the rectangular-pulse forcing j(t) = 0.04 s[sin(0.5t)]. Initial condition: x(0) = 0.5,y(0) =

44 M. Itoh & L. Chua (a) j(t) = 0 (b)j(t) = sin(0.5t) (c) j(t) = 0.04 s[sin(0.5t)] Fig. 41. Projection of the trajectories of Eq. (130) into the (ξ,η,ζ)-space via the coordinate transformation (171). Observe the difference among the three orbits. We can observe a gap in Figs. 41(b) and 41(c), which is not filled with trajectories. We set ω 0 =0.5, which satisfies Re[Z(iω 0 )] < 0. We deleted the transient trajectories to view the attractors clearly. The trajectory is coded using the rainbow color code for ζ, that is, the color evolves through violet, indigo, blue, green, yellow, orange and red, as ζ varies from its minimum to its maximum value. (a) A trajectory for j(t) = 0, which converges to a torus after the transient regime. Since the trajectory on the (x, y)-plane converges to a stable limit cycle as shown in Fig. 40(a), the corresponding trajectory in the (ξ, η, ζ)-space converges to a torus (see Appendix A for more details). The red portion of trajectory is not a periodic orbit, but a part of orbit on a converged torus. Initial condition: x(0) = 0.5,y(0) = 0.5. The initial point (x(0),y(0)) = (0.5, 0.5) for Eq. (130) is projected into the point (ξ(0),η(0),ζ(0)) = (10.5, 0, 0.5) via the coordinate transformation (171). (b) A chaotic trajectory under the sinusoidal forcing j(t) = sin(0.5t). Since Eq. (130) exhibits a chaotic oscillation on the (x, y)-plane as shown in Fig. 40(b), the corresponding trajectory in the (ξ,η,ζ)-space is also chaotic (see Appendix A for more details). Initial condition: x(0) = 0.5,y(0) = 0.5. (c) A chaotic trajectory under the rectangular-pulse forcing j(t) = 0.04 s[sin(0.5t)]. Since Eq. (130) exhibits a chaotic oscillation on the (x, y)-plane as shown in Fig. 40(c), the corresponding trajectory in the (ξ, η, ζ)-space is also chaotic (see Appendix A for more details). Initial condition: x(0) = 0.5,y(0) =

45 Chaotic Oscillation via Edge of Chaos Criteria (a) j(t) = sin(0.5t) (b) j(t) = 0.04 s[sin(0.5t)] Fig. 42. Poincaré maps of the forced FitzHugh Nagumo model defined by Eq. (130) with I = We set ω 0 =0.5, which satisfies Re[Z(iω 0 )] < 0. Many folding loci can be observed in the partial enlarged view of the Poincaré maps.(a) Poincaré map under the sinusoidal forcing j(t) = sin(0.5t), and its partial enlarged view (right). Initial condition: x(0) = 0.5,y(0) = 0.5. (b) Poincaré map under the rectangular-pulse forcing j(t) = 0.04 s[sin(0.5t)], and its partial enlarged view (right). Initial condition: x(0) = 0.5,y(0) =

46 M. Itoh & L. Chua Fig. 43. Coexistence of two attractors in the forced FitzHugh Nagumo model defined by Eq. (130) with I = We set ω 0 =0.5, which satisfies Re[Z(iω 0 )] < 0. A chaotic attractor (red) and a small periodic orbit (purple) coexist. The right figure gives a partially enlarged view. The sinusoidal forcing is given by j(t) = sin(0.5t). Initial condition for small periodic orbit: x(0) = 0.97,y(0) = Initial condition for chaotic attractor: x(0) = 0.5,y(0) = 0.5. Finally, let us study the case where Re[Z(iω 0 )] > 0, and we set ω 0 =0.18. Note that Eq. (130) can exhibit a chaotic oscillation, even if Re[Z(iω 0 )] > 0. We show the trajectories of Eq. (130) with ω 0 =0.18 in Fig. 45. In order to view their trajectories from a different perspective, let us project the trajectories of Eq. (130) into the (ξ,η,ζ)-space via the coordinate transformation: ξ(t) =(x(t) + 10) cos(0.18t), η(t) =(x(t) + 10) sin(0.18t), ζ(t) = y(t). (172) We show the projected trajectories in Fig. 46. Compare the orbits in Fig. 46 with those in Fig. 41. In Figs. 46(b) and 46(c), we can observe a gap, which is not filled with trajectories. We next show the corresponding Poincaré maps in Fig. 47. Compare the attractors in Fig. 47 with those in Fig. 42. Observe that if Re[Z(iω 0 )] < 0, the behavior of the chaotic attractors becomes more complex, that is, it has a dense set of folding loci. We conclude as follows: Folding loci for Re[Z(iω 0 )] < 0 Suppose that the angular frequency ω 0 of periodic forcing satisfies the edge of chaos condition (ii), that is, Re[Z(iω 0 )] < 0. Then, the chaotic attractor of the forced FitzHugh Nagumo model with I = and I =1.43 can exhibit a dense set of folding loci. (a) j(t) = sin(0.5t) (b) j(t) = 0.04 s[sin(0.5t)] Fig. 44. Average power P (t) of the forced FitzHugh Nagumo model defined by Eq. (130) with I = We set ω 0 =0.5, which satisfies Re[Z(iω 0 )] < 0. (a) P (t) < 0fort 0 under sinusoidal forcing. The average power from the forced signal is not dissipated in the chaotic one-port circuit. Here, the periodic forcing is given by j(t) = sin(0.5t). Initial condition: x(0) = 0.5,y(0) = 0.5. (b) P (t) > 0fort>0 under rectangular-pulse forcing. The average power delivered from the forced signal is dissipated in the one-port N. Here, the periodic forcing is given by j(t) = 0.04 s[sin(0.5t)]. Initial condition: x(0) = 0.5,y(0) =

47 Chaotic Oscillation via Edge of Chaos Criteria (a) j(t) = 0 (c) j(t) = 0.05 s[sin(0.18t)] (b)j(t) = sin(0.18t) Fig. 45. Trajectories of the forced FitzHugh Nagumo model defined by Eq. (130) with I = We set ω 0 =0.18, which satisfies Re[Z(iω 0 )] > 0. (a) Equation (130) with j(t) = 0 has an asymptotically-stable equilibrium point (blue), a stable limit cycle (red), and an unstable limit cycle (green) as its limit set. The coordinate of the asymptotically-stable equilibrium point (blue) is described by (x 0,y 0 ) (0.97, 2.087). In this case, Eq. (130) is an autonomous system. We calculate the unstable limit cycle with reverse time scaling. Initial condition for a stable limit cycle: x(0) = 0.5,y(0) = 0.5. Initial condition for an unstable limit cycle: x(0) = 0.97,y(0) = (b) A chaotic trajectory of Eq. (130) under the sinusoidal forcing j(t) = sin(0.18t). Initial condition: x(0) = 0.5, y(0) = 0.5. (c) A chaotic trajectory of Eq. (130) under the rectangular-pulse forcing j(t) = 0.05 s[sin(0.18t)]. Initial condition: x(0) = 0.5, y(0) =

48 M. Itoh & L. Chua (a) j(t) = 0 (b)j(t) = sin(0.18t) (c) j(t) = 0.05 s[sin(0.18t)] Fig. 46. Trajectories of Eq. (130) with I = , which are projected into the (ξ,η,ζ)-space via the coordinate transformation (172). We set ω 0 =0.18, which satisfies Re[Z(iω 0 )] > 0. Compare the trajectories in Fig. 46 with those in Fig. 41. Observe that the orbits in Figs. 46(b) and 46(c) are less dense than those in Figs. 41(b) and 41(c). The trajectory is coded using the rainbow color code for ζ, that is, the color evolves through violet, indigo, blue, green, yellow, orange and red, as ζ varies from its minimum to its maximum value. (a) A trajectory for j(t) = 0, which converges to a torus after the transient regime. Since the trajectory on the (x, y)-plane converges to a stable limit cycle as shown in Fig. 45(a), the corresponding trajectory in the (ξ, η, ζ)-space converges to a torus (see Appendix A for more details). We deleted the transient trajectory to view the attractor (torus) clearly. The red portion of trajectory is not a periodic orbit, but a part of orbit on a converged torus. Initial condition: x(0) = 0.5, y(0) = 0.5. The initial point (x(0),y(0)) = (0.5, 0.5) for Eq. (130) is projected into the point (ξ(0),η(0),ζ(0)) = (10.5, 0, 0.5) via the coordinate transformation (172). (b) A chaotic trajectory under the sinusoidal forcing j(t) = sin(0.18t). Since Eq. (130) exhibits a chaotic oscillation on the (x, y)-plane as shown in Fig. 45(b), the corresponding trajectory in the (ξ, η, ζ)-space is also chaotic (see Appendix A for more details). We deleted the transient trajectory to view the chaotic attractor clearly. The red portion of trajectory is not a periodic orbit, but a part of chaotic orbit. Initial condition: x(0) = 0.5, y(0) = 0.5. (c) A chaotic trajectory under the rectangular-pulse forcing j(t) = 0.05 s[sin(0.18t)]. Since Eq. (130) exhibits a chaotic oscillation on the (x, y)-plane as shown in Fig. 45(c), the corresponding trajectory in the (ξ,η,ζ)-space is also chaotic (see Appendix A for more details). We deleted the transient trajectory to view the chaotic attractor clearly. The red portion of trajectory is not a periodic orbit, but a part of chaotic orbit. Initial condition: x(0) = 0.5, y(0) =

49 Chaotic Oscillation via Edge of Chaos Criteria (a) j(t) = sin(0.18t) (b) j(t) = 0.05 s[sin(0.18t)] Fig. 47. Poincaré maps of Eq. (130) with I = We set ω 0 =0.18, which satisfies Re[Z(iω 0 )] > 0. Compare the attractors in Fig. 47 with those in Fig. 42. Observe that the folding loci in Fig. 47 are clearly separated from each other and are hence more visible than those in Fig. 42. (a) Poincaré map under the sinusoidal forcing j(t) = sin(0.18t). Initial condition: x(0) = 0.5,y(0) = 0.5. (b) Poincaré map under the rectangular-pulse forcing j(t) = 0.05 s[sin(0.18t)]. Initial condition: x(0) = 0.5,y(0) =

50 M. Itoh & L. Chua 5. Forced Van der Pol Oscillator Consider the forced Van der Pol oscillator defined by ẍ ɛ(1 x 2 )ẋ + x = A cos(ωt), (173) where ɛ>0isaparameter. This can be written as a system of first-order ordinary differential equations: ( x 3 ) ẋ = y ɛ 3 x, ẏ = x + j(t), (174) where j(t) =A cos(ωt). Equation (174) can be realized by the circuit shown in Fig. 48. Its dynamics can be given by where L 1 di 1 dt = v 1 ɛ ( i1 3 C 1 dv 1 dt = i 1 + j(t), 3 i 1 ), (175) C 1 =1, L 1 =1. (176) Substituting Eq. (176) into Eq. (175), we obtain ( 3 ) di 1 dt = v i1 1 ɛ 3 i 1, (177) dv 1 dt = i 1 + j(t). Equations (174) and (177) are identical, if we rename the variables i 1 = x, v 1 = y. (178) Let us study the equilibrium point of the following equation: ( ) x 3 ẋ = y ɛ 3 x, ẏ = x, (179) which does not have the forcing term. It has one equilibrium point at the origin: (x, y) =(0, 0). The Jacobian matrix J at this point is given by [ ɛ ] 1 J = 1 0. (180) The characteristic equation of J can be written as and its eigenvalues are given by λ 2 ɛλ +1=0, (181) λ = ɛ ± ɛ 2 4. (182) 2 The equilibrium point of Eq. (179), that is, the origin, is unstable, since λ 1 + λ 2 > 0andλ 1 λ 2 > 0. Here λ 1 and λ 2 are eigenvalues of Eq. (181). Let δx and δy denote infinitesimal variables in the neighborhood of the point (x 0,y 0 )=(0, 0), namely, x(t) =x 0 + δx(t) =δx(t), y(t) =y 0 + δy(t) =δy(t). (183) From Eq. (174), we obtain the variational equation d(δx) = δy + ɛδx, dt (184) d(δy) = δx + j(t). dt Applying the Laplace transform to each term in Eq. (184), we obtain Fig. 48. Circuit model for Eq. (174). Parameters: C 1 = 1,L 1 = 1. The nonlinear resistor is defined by v R = 3 «ir ɛ 3 i R. The independent current source j(t) denotes a periodic forcing. where sˆx(s) δx(0) = ŷ(s)+ ɛˆx(s), sŷ(s) δy(0) = ˆx(s)+ĵ(s), ˆx(s) = ŷ(s) = ĵ(s) = δx(t)e st dt, δy(t)e st dt, j(t)e st dt. (185) (186)

51 Chaotic Oscillation via Edge of Chaos Criteria Assuming zero initial condition δx(0) = 0 and δy(0) = 0, we obtain x(0) = x 0 =0, y(0) = y 0 =0. (187) Thus, the initial point is at the origin. Solving the first equation in Eq. (185) for ŷ(s), we obtain ˆx(s) = ŷ(s) s ɛ. (188) Substituting this equation into the second equation in Eq. (185), we obtain ( s + 1 ) ŷ(s) =ĵ(s). (189) s ɛ Consider next the impedance Z(s) defined by Z(s) = ŷ(s) ĵ(s) = Then Z(s) hasthetwopoles: p 1 = ɛ + ɛ s ɛ s 2 ɛs +1. (190), p 2 = ɛ ɛ 2 4. (191) 2 They are in the open right-half plane. Thus, the forced Van der Pol oscillator does not satisfy the edge of chaos condition (i). Furthermore, Eq. (184) does not have a small-signal periodic steady state since the equilibrium point (0, 0) is unstable. Consider next the small-signal circuit model for Eq. (184). The impedance Z(s) defined by Eq. (190) can be recast into Z(s) = 1 s + 1. (192) s ɛ Thus, the parameters in Fig. 2 are given by C =1, L =1, G =0, R = ɛ. (193) Since G = 0, the linear circuit Fig. 2 can be simplified to the circuit shown in Fig. 49. Observe that this circuit has a locally-active element R with a negative resistance equal to ɛ[ω]. Consider next Z(iω), which can be written as Z(iω) = iω ɛ 1 ω 2 iɛω = ɛ + iω(1 ω2 ɛ 2 ) (1 ω 2 ) 2 + ɛ 2 ω 2. (194) Fig. 49. Small-signal circuit model for Eq. (190) (left). The impedance of the one-port circuit connected across j s(t) is defined by Z(s) = L[v(t)] L[j = 1,where s(t)] 1 sc + sl + R C =1,L =1,R = ɛ <0. The independent current source j s(t) denotes a periodic forcing. Its real part and imaginary part are given respectively by ɛ Re[Z(iω)] = (1 ω 2 ɛ 2 ) 2 + ɛ 2 ω 2 < 0, Im[Z(iω)] = ω(1 ω2 ɛ 2 ) (1 ω 2 ) 2 +(ɛω) 2. (195) Since Re[Z(iω)] < 0, the edge of chaos condition (ii) is satisfied for all ω. However, we cannot calculate the following average power P n entering the one-port N for sufficiently large n, since the steady state of δy(t) does not exist because the right-half plane poles in Eq. (191) imply the transient component of the variational equation tends to infinity. P n = 1 T = 1 T nt (n 1)T nt (n 1)T p(τ)dτ δy(τ)j(t)dτ. (196) Here, δy(t) is defined to be the voltage v 1 [see Eqs. (178) and (183)]. Let us study the behavior of Eq. (174) with ɛ =8.53 [Kanamaru, 2007]. We first show the trajectory of Eq. (174) for j(t) = 0 in Fig. 50(a). In this case, the equilibrium point (origin) is unstable. We next show the trajectories of Eq. (174) with the periodic forcing j(t) 0 in Figs. 50(b) and 50(c). The periodic forcing signals are given by (see [Kanamaru, 2007]) and j(t) =1.2cos(0.2πt), (197) j(t) = s[cos(0.2πt)]. (198)

52 M. Itoh & L. Chua (a) j(t) =0 (b)j(t) =1.2cos(0.2πt) (c) j(t) = s[cos(0.2πt)] Fig. 50. Trajectories of the forced Van der Pol oscillator defined by Eq. (174) with ɛ =8.53, where Re[Z(iω)] < 0 for all ω. (a) A trajectory of Eq. (174) with j(t) = 0 tends to a stable limit cycle, and the origin is an unstable equilibrium point. A transient trajectory is shown in black. Observe that the transient regime tends to zero even though the variational equation had predicted that the transient will grow to infinity, because the variational equation is not applicable in this case, where the nonlinear term in the Van der Pol equation can no longer be neglected. In this case, Eq. (174) is an autonomous system. Initial condition: x(0) = 0.1,y(0) = 0.1. (b) A chaotic trajectory of Eq. (174) under sinusoidal forcing j(t) = 1.2cos(0.2πt). Initial condition: x(0) = 0.1,y(0) = 0.1. (c) A chaotic trajectory of Eq. (174) under rectangular-pulse forcing j(t) = s[cos(0.2πt)]. Initial condition: x(0) = 0.1,y(0) = 0.1. For cosine forcing, Eq. (174) exhibits a slow fast behavior and a chaotic oscillation. For rectangularpulse forcing, Eq. (174) exhibits a canard-like trajectory (see [Itoh & Murakami, 1994] about canard). In order to view their trajectories from a different perspective, let us project the trajectories into the (ξ,η,ζ)-space via the coordinate transformation: ξ(t) =(x(t) + 10) cos(0.2πt), η(t) =(x(t) + 10) sin(0.2πt), ζ(t) = y(t). (199) Observe that the three trajectories in Fig. 51 are quite different. If j(t) = 0, the trajectory moves on the two-dimensional manifold (torus) in the (ξ,η,ζ)-space after a transient behavior, as shown in Fig. 51(a). However, for periodic forcing, the projected trajectories in the (ξ,η,ζ)-space, as shown in Figs. 51(b) and 51(c), are much more complicated. We also show their Poincaré maps in Fig. 52. The chaotic attractors seem to be compressed tightly, and they look like a one-dimensional curve

53 Chaotic Oscillation via Edge of Chaos Criteria (a) j(t) =0 (b)j(t) =1.2cos(0.2πt) (c) j(t) = s[cos(0.2πt)] Fig. 51. Projection of the trajectories of Eq. (174) into the (ξ,η,ζ)-space via the coordinate transformation (199). Observe that the three trajectories are quite different. Recall that Re[Z(iω)] < 0 for all ω. The trajectory is coded using the rainbow color code for ζ, that is, the color evolves through violet, indigo, blue, green, yellow, orange and red, as ζ varies from its minimum to its maximum value. (a) A trajectory for j(t) = 0, which converges to a torus with rectangular-shaped cross-section after the transient regime. Since the trajectory on the (x, y)-plane converges to a stable limit cycle as shown in Fig. 50(a), the corresponding trajectory in the (ξ, η, ζ)-space converges to a torus after the transient regime (see Appendix A for more details). We deleted the transient trajectory to view the attractor (torus) clearly. Initial condition: x(0) = 0.1,y(0) = 0.1. The initial point (x(0),y(0)) = (0.1, 0.1) for Eq. (174) is projected into the point (ξ(0),η(0),ζ(0)) = (10.1, 0, 0.1) via the coordinate transformation (199). (b) A chaotic trajectory under the periodic forcing j(t) =1.2cos(0.2πt). Since Eq. (174) exhibits a chaotic oscillation on the (x, y)-plane as shown in Fig. 50(b), the corresponding trajectory in the (ξ,η,ζ)-space is also chaotic (see Appendix A for more details). We deleted the transient trajectory to view the chaotic attractor clearly. Initial condition: x(0) = 0.1,y(0) = 0.1. (c) A chaotic trajectory under the rectangular-pulse forcing j(t) = s[cos(0.2πt)]. Since Eq. (174) exhibits a chaotic oscillation on the (x, y)-plane as shown in Fig. 50(c), the corresponding trajectory in the (ξ,η,ζ)-space is also chaotic (see Appendix A for more details). We deleted the transient trajectory to view the chaotic attractor clearly. Initial condition: x(0) = 0.1,y(0) =

54 M. Itoh & L. Chua (a) j(t) = 1.2cos(0.2πt) (b) j(t) = s[cos(0.2πt)] Fig. 52. Poincaré maps of the forced Van der Pol oscillator defined by Eq. (174) with ɛ =8.53. Recall that Re[Z(iω)] < 0 for all ω. The chaotic attractors seem to be compressed tightly, and they look like a one-dimensional curve. (a) Poincaré map under the sinusoidal forcing j(t) = 1.2cos(0.2πt). Initial condition: x(0) = 0.1,y(0) = 0.1. (b) Poincaré map under the rectangular-pulse forcing j(t) = s[cos(0.2πt)]. Initial condition: x(0) = 0.1,y(0) = 0.1. We conclude as follows: Chaotic oscillation of Eq. (174) For cosine forcing, Eq. (174) exhibits a chaotic oscillation with slow fast behavior. For rectangular-pulse forcing, Eq. (174) exhibits a canard-like trajectory. The chaotic attractors appeared to be tightly compressed. Note that if we do not know the parameter given in [Kanamaru, 2007], we cannot obtain their values from the edge of chaos condition (ii) because the edge of chaos condition (i) is not satisfied. It is known that chaos takes place in a narrow range of ɛ. Recall that y(t) andj(t) are equivalent to the voltage v 1 (t) across the capacitor C 1 and the current j(t) of the independent current source, respectively [see Fig. 48 and Eqs. (175) and (178)]. That is, y(t) andj(t) are equivalent to the terminal voltage and current of the independent current source in Fig. 48, respectively. Thus, we can define the instantaneous power entering the one-port N at time τ by y(τ)j(τ), and the average power P (t) by P (t) = 1 t y(τ)j(τ)dτ. (200) t 0 (a) j(t) = 1.2cos(0.2πt) (b) j(t) = s[cos(0.2πt)] Fig. 53. Average power P (t) of the forced Van der Pol oscillator defined by Eq. (174). Recall that Re[Z(iω)] < 0 for all ω. P (t) becomes positive, even if time t is sufficiently large. That is, the forced Van der Pol one-port N defined by Eq. (174) dissipates energy delivered by the current source j(t) for all time t>50. Here, the periodic forcing signals in Eq. (174) are given respectively by j(t) = 1.2cos(0.2πt) and j(t) = s[cos(0.2πt)]

55 Chaotic Oscillation via Edge of Chaos Criteria We show the average power P (t) in Fig. 53, which is obtained from the chaotic solution of Eq. (174). Observe that the average power P (t) enteringthe one-port N is positive, even if Re[Z(iω)] < 0 for all ω, because Eq. (174) is not a linear equation, that is, it includes the nonlinear term ɛx3 3. where Average power P (t) of Eq. (174) a n (V )= The forced Van der Pol oscillator (174) dissipates average power in the one-port N delivered from the forced periodic signal, even if Re[Z(iω)] < 0 for all ω. 6. Hodgkin Huxley and Morris Lecar Models 6.1. Hodgkin Huxley model The Hodgkin Huxley model is a mathematical model for simulating the behavior of nerve cells in the squid giant axon. Its equivalent circuit model isshowninfig.54of[hodgkin&huxley, 1952]. This model is written as a set of four nonlinear differential equations [Hodgkin & Huxley, 1952; Chua et al., 2012; Chua, 2013]: C M dv dt = g Nam 3 h(v V Na ) g K n 4 (V V K ) g L (V V L )+I, dn dt = a n(v )(1 n) b n (V )n, a m (V )= dm dt = a m(v )(1 m) b m (V )m, dh dt = a h(v )(1 h) b h (V )h, 0.01(V + 10) e 0.1(V +10) 1, b n(v )=0.125e V 80, 0.1(V + 25) e 0.1(V +25) 1, b m(v )=4e V 18, a h (V )=0.07e 0.05V, b h (V )= (201) 1 e 0.1(V +30) +1. (202) Here, C M is the membrane capacitance, V = E E r is the displacement of the membrane potential from its resting value, E is the membrane capacitor voltage, E r is the absolute value of the resting potential, I is the total current through the membrane, are the potassium, sodium, and leak conductances, respectively. Furthermore, V Na = E Na E r, V K = E K E r, V L = E L E r are the displacements from its resting value, where E Na and E K are the sodium and potassium ion batteries, respectively, and E L are the leakage battery voltage. The variables m, n, andh are state variables defining the sodium and potassium ion channel memristors [Chua et al., 2012]. In this paper, the following parameters are used: g K = 1 R K, g Na = 1 R Na,andg L = 1 R L V Na = 115, V k =12, V L = , g K =36, g Ca = 120, g L =0.3, C M =1, (203) where all physical units are omitted for simplicity. Equation (201) can be realized by the memristive Hodgkin Huxley model [Chua et al., 2012; Chua, 2013]. Consider next the forced Hodgkin Huxley equations defined by Fig. 54. Hodgkin Huxley circuit model. I is the total current through the membrane. R Na and R K are incorrectly identified as time-varying resistances in [Hodgkin & Huxley, 1952], but are now understood as sodium ion channel memristor, and potassium ion channel memristor, respectively [Chua et al., 2012]. The other components are constants. C M dv dt = g Nam 3 h(v V Na ) g K n 4 (V V K ) g L (V V L )+I + j(t),

56 M. Itoh & L. Chua Fig. 55. Memristive circuit model for Eq. (204). Two current sources I and j(t) are added. Here, E, E Na, E k,ande L in Fig. 54 are replaced with V, V Na, V k,andv L, respectively. dn dt = a n(v )(1 n) b n (V )n, dm dt = a m(v )(1 m) b m (V )m, dh dt = a h(v )(1 h) b h (V )h, (204) where j(t) denotes the periodic forcing given by j(t) =A sin (ωt). The forced Hodgkin Huxley equations (204) can be realized by the circuit in Fig. 55, where I and j(t) are realized by two current sources. Let Q = (V 0,n 0,m 0,h 0 ) be the equilibrium point of Eq. (201) with external DC membrane current I = I 0.LetδV, δn, δm, δh, denote infinitesimal variables in the neighborhood of the equilibrium point Q, namely, V (t) =V 0 + δv (t), n(t) =n 0 + δn(t), m(t) =m 0 + δm(t), h(t) =h 0 + δh(t). (205) If we substitute Eq. (205) into Eq. (204), we can obtain the variational equation. Applying the Laplace transform to this variational equation, we can obtain the admittance defined by Y (s) = ĵ(s) ˆV (s), (206) where ˆV (s) = 0 δv (t)e st dt, ĵ(s) = 0 j(t)e st dt. (207) In this paper, we use the admittance Y (s) whichis obtained by using the small-signal Hodgkin Huxley model about a DC equilibrium point [Chua et al., 2012; Chua, 2013]. The admittance Y (s) hasthe form Y (s) = b 4s 4 + b 3 s 3 + b 2 s 2 + b 1 s + b 0 a 3 s 3 + a 2 s 2 + a 1 s + a 0, (208) where the parameters a 0,a 1,a 2,a 3,b 0,b 1,b 2,b 3,b 4 are calculated by explicit formulas derived in [Chua et al., 2012]. Note that Y (s) =Y (s) for the same DC equilibrium point Q =(V 0,n 0,m 0,h 0 ). The edge of chaos criteria for the admittance Y (s) is given by [Chua, 2013]: Hodgkin Huxley Edge of Chaos Criteria Condition (i) All zeros of Y (s) are on the left-half plane. Condition (ii) Re[Y (iω)] < 0 for at least one frequency ω = ω 0. By applying the above criteria to Eq. (208), we can obtain the edge of chaos regime I (209) For more details, see [Chua et al., 2012; Chua, 2013]

57 Chaotic Oscillation via Edge of Chaos Criteria (a) j(t) =0 (b)j(t) =1.6sin(0.5t) (c) j(t) = 2.25 s[sin(0.5t)] Fig. 56. Trajectories of the forced Hodgkin Huxley model defined by Eq. (204) with I = 9.7. We set ω 0 =0.5, which satisfies Re[Y (iω 0 )] < 0. (a) Equation (204) with j(t) = 0 has an asymptotically-stable equilibrium point (red) and a large stable limit cycle (purple) with initial condition V (0) = 5.3,n(0) = 0.4,m(0) = 0.1,h(0) = 0.1. The coordinate of the asymptotically-stable equilibrium point (red) is described by (V 0,n 0,m 0,h 0 ) ( 5.318, , , ). In this case, Eq. (204) is an autonomous system. (b) A chaotic trajectory of Eq. (204) under the sinusoidal forcing j(t) = 1.6sin(0.5t). Initial condition: V (0) = 5.3,n(0) = 0.4,m(0) = 0.1,h(0) = 0.1. (c) A chaotic trajectory of Eq. (204) under the rectangular-pulse forcing j(t) = 2.25 s[sin(0.5t)]. Initial condition: V (0) = 5.3,n(0) = 0.4,m(0) = 0.1,h(0) = 0.1. We show the behavior of Eq. (204) without forcing in Fig. 56(a). In this case, Eq. (204) is an autonomous system. We found an asymptoticallystable equilibrium point (red) and a large stable limit cycle (purple) as its attractors. There may be some unstable closed orbits as shown in [Guckenheimer & Oliva, 2002]. The coordinate of the asymptotically-stable equilibrium point (red) is described by (V 0,n 0,m 0,h 0 ) ( 5.318, , , ). (210)

58 M. Itoh & L. Chua In Fig. 56(a), we deleted the transient trajectories to view the equilibrium point clearly. Let us choose ω 0 =0.5, which satisfies the edge of chaos condition (ii), that is, Re[Y (iω 0 )] < 0. They can be obtained from Fig. 19 in [Chua, 2013]. We next show the trajectories of Eq. (204) under the periodic forcing j(t) 0 in Figs. 56(b) and 56(c). In this case, Eq. (204) is a nonautonomous system. The periodic forcing signals are given by the sinusoidal function j(t) =1.6sin(0.5t), (211) and the rectangular-pulse function j(t) = 2.25 s[sin(0.5t)]. (212) Observe that Eq. (204) exhibits a chaotic oscillation under periodic forcing. In order to view their trajectories from a different perspective, let us project the trajectories into the (ξ,η,ζ)-space via the transformation ξ(t) =(V (t) + 200) cos(0.5t), η(t) =(V (t) + 200) sin(0.5t), ζ(t) = n(t) 2 + m(t) 2 + h(t) 2. (213) We show their trajectories in Fig. 57. Observe that Eq. (204) exhibits an interesting chaotic attractor. We also show the Poincaré maps in Fig. 58. In order to plot the points (V [kt],n[kt],m[kt],h[kt]), (214) (k = 1, 2,...) in the three-dimensional space, we used a 3D bubble chart 12 to draw 3D bubbles (solid spheres) at positions (V [kt],n[kt],m[kt]) with sizes h(kt), which are filled with different colors. Here, T = 2π = 2π is the period of the forcing signal. Observe that the Poincaré maps exhibit ω a slightly complicated behavior at the bottom corner. We also observed the coexistence of two periodic orbits as shown in Fig. 59, when the sinusoidal forcing is given by j(t) = 0.1sin(0.5t). These periodic motions bifurcate to a chaotic attractor if A is increased. Thus, we conclude as follows: Oscillation in the forced Hodgkin Huxley model With a small periodic forcing, Eq. (204) with I = 9.7 exhibits a small periodic response. If we increase the amplitude of the periodic forcing, it can evolve to a chaotic attractor. Since V (t) andj(t) are assumed to be the terminal voltage and current of the independent current source, respectively (see Fig. 55), we can define the instantaneous power entering the one-port N at time τ by V (τ)j(τ) and the average power P (t) by P (t) = 1 t t 0 V (τ)j(τ)dτ. (215) We show the average power P (t) in Fig. 60, where the periodic forcing signals are given by j(t) = 1.6 sin(0.5t) and j(t) = 2.25 s[sin(0.5t)]. Observe that the average power P (t) entering the one-port N is negative if time t is large. Thus, we conclude as follows 13 : Average power for Re[Y (iω 0 )] < 0 The forced Hodgkin Huxley model (204) with I = 9.7 does not dissipate average power from the current source with j(t) = 1.6 sin(0.5t), and j(t) = 2.25 s[sin(0.5t)]. That is, the one-port N connected across the current source can generate energy by chaotic oscillation, in view of the presence of at least one locally-active circuit element (e.g. the sodium channel conductance) inside N. We next study chaotic behavior for the periodic forcing signals which satisfy Re[Y (iω 0 )] > 0. Let us choose ω 0 = 2 (see Fig. 19 in [Chua, 2013]). We show the trajectories of Eq. (204) in Fig. 61. Here, the periodic forcing signals are given by the sinusoidal function: j(t) =1.6sin(2t), (216) and the rectangular-pulse function j(t) = 3.3 s[sin(2t)]. (217) 12 The 3D bubble chart is Mathematica s data visualization tool. It is used to visualize four-dimensional data sets. The first three data are used to place a bubble (solid sphere) in the (x, y, z)-space, and the last data is used to determine the size of the bubble (solid sphere). A 3D bubble chart can compare the relationships between data objects in four dimensions. 13 We obtain similar results with the periodic forcing signals j(t) =1.6sin(2t) andj(t) =3.3s[sin(2t)]. Here, the angular frequency ω 0 = 2 satisfies Re[Y (iω 0 )] >

59 Chaotic Oscillation via Edge of Chaos Criteria (a) j(t) =0 (c) j(t) = 2.25 s[sin(0.5t)] (b)j(t) =1.6sin(0.5t) Fig. 57. Projection of the trajectories of Eq. (204) into the (ξ,η,ζ)-space via the coordinate transformation (213). We set ω 0 = 0.5, which satisfies Re[Y (iω 0 )] < 0. Observe that Eq. (204) can exhibit a chaotic behavior by periodic forcing. The trajectory is coded using the rainbow color code for ζ, that is, the color evolves through violet, indigo, blue, green, yellow, orange and red, as ζ varies from its minimum to its maximum value. (a) A trajectory of Eq. (204) with j(t) = 0, which converges to the torus after the transient regime. Since the trajectory in the (V,n, m,h)-space converges to a large stable limit cycle as shown in Fig. 56(a), the corresponding trajectory in the (ξ,η,ζ)-space converges to a torus (see Appendix A for more details). We deleted the transient trajectory to view the attractor (torus) clearly. Initial condition: V (0) = 5.3,n(0) = 0.4,m(0) = 0.1,h(0) = 0.1. The initial point (V (0),n(0),m(0),h(0)) = ( 5.3, 0.4, 0.1, 0.1) for Eq. (204) is projected into the point (ξ(0),η(0),ζ(0)) = (194.7, 0, 0.18) (194.7, 0, 0.424) via the coordinate transformation (213). (b) A chaotic trajectory of Eq. (204) under the sinusoidal forcing j(t) = 1.6sin(0.5t). Since Eq. (204) exhibits a chaotic oscillation in the (V,n, m,h)-space as shown in Fig. 56(b), the corresponding trajectory in the (ξ,η,ζ)-space is also chaotic (see Appendix A for more details). We deleted the transient trajectory to view the chaotic attractor clearly. Initial condition: V (0) = 5.3,n(0) = 0.4,m(0) = 0.1,h(0) = 0.1. (c) A chaotic trajectory of Eq. (204) under the rectangular-pulse forcing j(t) = 2.25 s[sin(0.5t)]. Since Eq. (204) exhibits a chaotic oscillation in the (V,n, m,h)-space as shown in Fig. 56(c), the corresponding trajectory in the (ξ, η, ζ)-space is also chaotic (see Appendix A for more details). We deleted the transient trajectory to view the chaotic attractor clearly. Initial condition: V (0) = 5.3,n(0) = 0.4,m(0) = 0.1,h(0) =

60 M. Itoh & L. Chua (a) j(t) = 1.6sin(0.5t) (b) j(t) = 2.25 s[sin(0.5t)] Fig. 58. Poincaré maps of the forced Hodgkin Huxley model defined by Eq. (204) with I = 9.7. We set ω 0 =0.5, which satisfies Re[Y (iω 0 )] < 0. We used a 3D bubble chart to draw 3D bubbles (solid spheres) at positions (V [kt],n[kt],m[kt]) with sizes h(kt), which are filled with different colors. Observe that the Poincaré maps exhibit a slightly complicated behavior in the bottom corner. (a) Poincaré map under the sinusoidal forcing j(t) = 1.6sin(0.5t). Initial condition: V (0) = 5.3,n(0) = 0.4,m(0) = 0.1,h(0) = 0.1. (b) Poincaré map under the rectangular-pulse forcing j(t) = 2.25 s[sin(0.5t)]. Initial condition: V (0) = 5.3,n(0) = 0.4,m(0) = 0.1,h(0) = 0.1. Fig. 59. Coexistence of two periodic responses of Eq. (204) with I = 9.7 when driven by a small sinusoidal signal j(t) =0.1sin(0.5t), where ω 0 =0.5 andre[y (iω 0 )] < 0. A large periodic orbit (purple) and a small periodic orbit (red) coexist. An enlargement of the tiny square on the bottom of the large periodic orbit is shown on the right side of this figure. Initial condition for large periodic orbit: V (0) = 5.3,n(0) = 0.4,m(0) = 0.4,h(0) = 0.4. Initial condition for small periodic orbit: V (0) = 5.3,n(0) = 0.4,m(0) = 0.1,h(0) =

61 Chaotic Oscillation via Edge of Chaos Criteria (a) j(t) =1.6sin(0.5t) (b) j(t) =3s[sin(0.5t)] Fig. 60. Average power P (t) of the forced Hodgkin Huxley model defined by Eq. (204) with I = 9.7. Here, we set ω 0 =0.5, which satisfies Re[Y (iω 0 )] < 0. (a) P (t) < 0fort>0 under the sinusoidal forcing. The average power from the forced signal is not dissipated in the one-port N connected across the forcing current source j(t). Here, the periodic forcing is given by j(t) = 1.6sin(0.5t). Initial condition: V (0) = 5.3,n(0) = 0.4,m(0) = 0.1,h(0) = 0.1. (b) P (t) < 0 for t > 0 under the rectangular-pulse forcing. The average power from the forced signal is also not dissipated in the one-port N. Here, the periodic forcing is given by j(t) = 3 s[sin(0.5t)]. Initial condition: V (0) = 5.3,n(0) = 0.4,m(0) = 0.1,h(0) = 0.1. Note that the trajectories Figs. 61(b) and 61(c) at first move around the small neighborhood of the red point: (V 0,n 0,m 0,h 0 ) ( 5.318, , , ) (218) shown in Fig. 61(a). However, they eventually leave this point, and exhibit torus-like trajectories as shown in Fig. 62. Compare the two trajectories in Fig. 62. We also show the trajectories projected into the (ξ,η,ζ)-space in Fig. 63. Here, the coordinate transformation is given by ξ(t) =(V (t) + 200) cos(2t), η(t) =(V (t) + 200) sin(2t), (219) ζ(t) = n(t) 2 + m(t) 2 + h(t) 2. Observe that all trajectories move on the surface of atorus. We next show their Poincaré maps in Fig. 64. From these Poincaré maps, we conclude that the attractors form a torus. Thus, we conclude as follows: Choose ω 0 =0.2, which satisfies Re[Z(iω 0 )] > 0. Then, the forced Hodgkin Huxley model (204) with I = 9.7 forms a torus in the (ξ,η,ζ)-space when driven by the periodic forcing signals j(t) = 1.6sin(2t) and j(t) = 3.3 s[sin(2t)] Morris Lecar model The Morris Lecar model describes a barnacle giant muscle fiber. We show its equivalent circuit model in Fig. 65 [Morris & Lecar, 1981]. This model is defined by a set of three differential equations [Morris & Lecar, 1981; Sah et al., 2016]: C m dv dt = g CaM(V E Ca ) g K N(V E K ) g L (V E L )+I, dm = λ M (V )(M (V ) M), dt dn dt = λ N(V )(N (V ) N), where { [ V +1.2 M (V )=0.5 1+tanh 18 [ ] V +1.2 λ M (V )=0.8cosh, 36 { [ V 2 N (V )=0.5 1+tanh 30 [ ] V 2 λ N (V )=0.04 cosh. 60 ]}, ]}, (220) (221) Here, C m is the membrane capacitance, V is the membrane capacitor voltage, I is the external membrane current, g K and g Ca are the potassium and

62 M. Itoh & L. Chua (a) j(t) =0 (b)j(t) =1.6sin(2t) (c) j(t) = 3.3 s[sin(2t)] Fig. 61. Trajectories of the forced Hodgkin Huxley model defined by Eq. (204) with I = 9.7. We set ω 0 = 2, which satisfies Re[Z(iω 0 )] > 0. Compare the three trajectories in Fig. 61 with those in Fig. 56. The trajectories in Figs. 61(b) and 61(c) at first move around a small neighborhood of the red point: (V 0,n 0,m 0,h 0 ) in Fig. 61(a). However, they eventually leave this point and exhibit a torus-like behavior [see Fig. 62(b)]. On the other hand, the trajectories in Figs. 56(b) and 56(c) exhibit a chaotic behavior. (a) Equation (204) with j(t) = 0 has an asymptotically-stable equilibrium point (red) and a stable limit cycle (purple) with initial condition V (0) = 5.3,n(0) = 0.4,m(0) = 0.1,h(0) = 0.1. The coordinate of the asymptotically-stable equilibrium point (red) is given by (V 0,n 0,m 0,h 0 ) ( 5.318, , , ). In this case, Eq. (204) is an autonomous system. (b) An attractor of Eq. (204) under the sinusoidal forcing j(t) = 1.6sin(2t). Initial condition: V (0) = 5.3,n(0) = 0.4,m(0) = 0.1,h(0) = 0.1. (c) An attractor of Eq. (204) under the rectangular-pulse forcing j(t) = 3.3 s[sin(2t)]. Initial condition: V (0) = 5.3,n(0) = 0.4,m(0) = 0.1,h(0) =

63 Chaotic Oscillation via Edge of Chaos Criteria (a) ω 0 =0.5, Re[Z(iω 0 )] < 0 (b) ω 0 =2,Re[Z(iω 0 )] > 0 Fig. 62. Partial enlarged views of the trajectories of Eq. (204), which are moving in a neighborhood of the point (V 0,n 0,m 0,h 0 ) ( 5.318, , , ) (marked red). (a) A chaotic trajectory under the sinusoidal forcing j(t) =1.6sin(0.5t), which satisfies Re[Z(iω 0 )] < 0, where ω 0 =0.5. (b) A torus-like trajectory under the sinusoidal forcing j(t) =1.6sin(2t), which satisfies Re[Z(iω 0 )] > 0, where ω 0 = 2. Initial condition for (a) and (b): V (0) = 5.3,n(0) = 0.4,m(0) = 0.1,h(0) = 0.1. calcium conductances, respectively, E Ca and E K denote the battery voltages associated with the calcium ion, and the potassium ion, and g L and E L denote the leakage conductance and the leakage battery voltage, respectively. The variables M and N are analogous to m and n in Eq. (201). In this paper, the following parameters are used: E Ca = 120, E k = 84, E l = 60, g K =8, g Ca =4.4, g L =2.0, C m =20.0, (222) where all physical units are omitted for simplicity. The Morris Lecar model (220) can be realized by the memristive Morris Lecar circuit shown in Fig. 12 of [Sah et al., 2016]. Consider the forced Morris Lecar equations defined by C m dv dt = g CaMh(V E Ca ) g K N(V E K ) g L (V E L ) + I + j(t), dm = λ M (V )(M (V ) M), dt dn dt = λ N(V )(N (V ) N), (223) where j(t) denotes the periodic forcing given by j(t) = A sin(ωt). The forced Morris Lecar equations can be realized by the circuit in Fig. 66, where I and j(t) are realized by two current sources. Similarly, we can obtain the admittance Y (s)to calculate the edge of chaos criteria. The admittance has the form Y (s) = b 2s 2 + b 1 s + b 0, (224) a 1 s + a 0 where a 0,a 1 and b 0,b 1,b 2 are constants. The detailed calculation of the admittance Y (s) isgiven in [Sah et al., 2016]. The two edge of chaos regimes for the forced Morris Lecar model are also given in [Sah et al., 2016] <I< <I< (225)

64 M. Itoh & L. Chua (a) j(t) =0 (b)j(t) =1.6sin(2t) (c) j(t) = 3.3 s[sin(2t)] Fig. 63. Projection of the trajectories of Eq. (204) into the (ξ,η,ζ)-space via the coordinate transformation (219). We set ω 0 = 2, which satisfies Re[Z(iω 0 )] > 0. Observe that all trajectories exhibit a torus after the transient regime. We deleted the transient trajectories to view the attractors (toruses) clearly. The trajectory is coded using the rainbow color code for ζ, that is, the color evolves through violet, indigo, blue, green, yellow, orange and red, as ζ varies from its minimum to its maximum value. (a) A trajectory of Eq. (204) with j(t) = 0. Since the trajectory in the (V,n,m,h)-space converges to a stable limit cycle as shown in Fig. 62(a), the corresponding trajectory in the (ξ,η, ζ)-space converges to a torus (see Appendix A for more details). Initial condition: V (0) = 5.3,n(0) = 0.4,m(0) = 0.1,h(0) = 0.1. The initial point (V (0),n(0),m(0),h(0)) = ( 5.3, 0.4, 0.1, 0.1) for Eq. (204) is projected into the point (ξ(0),η(0),ζ(0)) = (194.7, 0, 0.18) (194.7, 0, 0.424) via the coordinate transformation (219). (b) A trajectory of Eq. (204) under the sinusoidal forcing j(t) =1.6sin(2t). Since the trajectory convergestoatorusinthe(v,n, m,h)-space as shown in Fig. 62(b), the corresponding trajectory in the (ξ,η,ζ)-space also converges to a torus. Initial condition: V (0) = 5.3,n(0) = 0.4,m(0) = 0.1,h(0) = 0.1. (c) A trajectory of Eq. (204) under the rectangular-pulse forcing j(t) = 3.3 s[sin(2t)]. Since the trajectory converges to a torus in the (V,n, m,h)-space as shown in Fig. 62(c), the corresponding trajectory in the (ξ,η, ζ)-space also converges to a torus after the transient regime. Initial condition: V (0) = 5.3,n(0) = 0.4,m(0) = 0.1,h(0) =

65 Chaotic Oscillation via Edge of Chaos Criteria (a) j(t) = 1.6 sin(2t) (b) j(t) = 3.3 s[sin(2t)] Fig. 64. Poincaré maps of the forced Hodgkin Huxley model defined by Eq. (204) with I = 9.7. We set ω 0 =2,which satisfies Re[Z(iω 0 )] > 0. We used 3D bubble chart to draw 3D bubbles (solid spheres) at positions (V [kt],n[kt],m[kt]) with sizes h(kt), which are filled with different colors. Observe that the attractors behave like a torus. (a) Poincaré map of Eq. (130) under the sinusoidal forcing j(t) = 1.6sin(2t). Initial condition: V (0) = 5.3,n(0) = 0.4,m(0) = 0.1,h(0) = 0.1. (b) Poincaré map of Eq. (130) under the rectangular-pulse forcing j(t) = 3.3 s[sin(2t)]. Initial condition: V (0) = 5.3,n(0) = 0.4,m(0) = 0.1,h(0) = 0.1. Fig. 65. Morris Lecar circuit model. g ca and g K are time-varying conductances. The other components are constants. I is an applied current. Fig. 66. Circuit model for Eq. (223). Two current sources I and j(t) are added

66 M. Itoh & L. Chua (a) j(t) =0 (b)j(t) =21sin(0.05t) (c) j(t) = 15.4 s[sin(0.05t)] Fig. 67. Trajectories of the forced Morris Lecar model defined by Eq. (223) with I = We set ω 0 =0.05, which satisfies Re[Y (iω 0 )] < 0. (a) A trajectory of Eq. (223) tends to an asymptotically-stable equilibrium point as t if j(t) =0. The coordinate of the asymptotically-stable equilibrium point is given by (V 0,M 0,N 0 ) (7.799, , ). In this case, Eq. (223) is an autonomous system. Initial condition: V (0) = 0.1,M(0) = 0.1,N(0) = 0.1. (b) A chaotic trajectory of Eq. (223) under the sinusoidal forcing j(t) = 21sin(0.05t). Initial condition: V (0) = 0.1,M(0) = 0.1,N(0) = 0.1. (c) A chaotic trajectory of Eq. (223) under the rectangular-pulse forcing j(t) = 15.4 s[sin(0.05t)]. Initial condition: V (0) = 0.1,M(0) = 0.1,N(0) =

67 Let us choose I = , which satisfies Eq. (225). We first show the trajectory of Eq. (223) with j(t) = 0 in Fig. 67(a). In this case, Eq. (223) is an autonomous system. A trajectory tends to an asymptotically-stable equilibrium point: (V 0,M 0,N 0 ) (7.799, , ), (226) as t. In order to apply the forcing signals, let us choose ω 0 =0.05, which satisfies the edge of chaos condition (ii), that is, ω 0 satisfies Re[Y (iω 0 )] < 0. They can be obtained from Fig. 21 in [Sah et al., 2016]. We show the trajectories of Eq. (223) under the periodic forcing j(t) 0 in Figs. 67(b) and 67(c). Here, the periodic forcing signals are given respectively by the sinusoidal function j(t) =21sin(0.05t), (227) and the rectangular-pulse function j(t) = 15.4 s[sin(0.05t)]. (228) Observe that Eq. (223) exhibits chaotic oscillations under periodic forcing. In order to view their trajectories from a different perspective, let us project the trajectories into the (ξ,η,ζ)-space via the transformation ξ(t) =(V (t) + 20) cos(0.05t), η(t) =(V (t) + 20) sin(0.05t), ζ(t) = M(t) 2 + N(t) 2. (229) We show their trajectories in Fig. 68. We also show the corresponding Poincaré maps in Fig. 69. We plotted the points (V (nt ),M(nT ),N(nT )) (n = 1, 2,...)inthe(V,M,N)-space, where T = 2π = ω 0 2π denotes the period of the forcing signal. We 0.05 can observe some folding loci. Furthermore, the attractors on the Poincaré maps seem to be tightly compressed, and look like a one-dimensional curve. From our computer simulations, we obtain the following result: Chaotic Oscillation via Edge of Chaos Criteria Oscillation in the forced Morris Lecar model (223) Under small periodic forcing, Eq. (223) with I = exhibits a small periodic response. If we increase the amplitude of the periodic forcing further, the periodic response grows, and evolves to a chaotic attractor. Since V (t)andj(t) are the terminal voltage and current of the independent current source, respectively (see Fig. 66), we can define the instantaneous power entering the one-port N at time τ by V (τ)j(τ) and the average power P (t) by P (t) = 1 t t 0 V (τ)j(τ)dτ. (230) We show the average power P (t) for Eq. (223) in Fig. 70, where the periodic forcing is given by j(t) =21sin(0.05t). The average power P (t) entering the one-port N is negative if time t is large. However, in the case of the rectangular-pulse j(t) = 15.4 s[sin(0.05t)], the average power P (t) becomes positive. Thus, we conclude as follows 14 : Average power for Re[Y (iω 0 )] < 0 Choose ω 0 =0.05 which satisfies Re[Y (iω 0 )] < 0. Then, we obtain the following result: sinusoidal forcing The forced Morris Lecar model (223) does not dissipate average power in the oneport N connected across the current source j(t) = 21sin(0.05t). That is, the one-port N generates energy by chaotic oscillation, powered by a locally-active element in N. rectangular-pulse forcing The forced Morris Lecar model (223) dissipates average power delivered from the rectangular-pulse current source j(t)= 15.4 s[sin(0.05t)]. Finally, let us study the case where the angular frequency ω 0 of the forced signal satisfies Re[Y (iω 0 )] > 0. We set ω 0 =0.2 (seefig.21in 14 We obtain similar results with the periodic forcing signals j(t) =20sin(0.2t) andj(t) =24.1s[sin(0.2t)]. Here, the angular frequency ω 0 =0.2 satisfyre[y (iω 0 )] >

68 M. Itoh & L. Chua (a) j(t) =0 (c) j(t) = 15.4 s[sin(0.05t)] (b)j(t) =21sin(0.05t) Fig. 68. Projection of the trajectories of Eq. (223) with I = into the (ξ,η,ζ)-space via the coordinate transformation (229). We set ω 0 =0.05, which satisfies Re[Y (iω 0 )] < 0. The trajectory is coded using the rainbow color code for ζ, that is, the color evolves through violet, indigo, blue, green, yellow, orange and red, as ζ varies from its minimum to its maximum value. (a) A trajectory of Eq. (223) with j(t) = 0, which converges to a thick orange circle after the transient regime. Since the trajectory in the (V,M,N)-space tends to an asymptotically-stable equilibrium point as shown in Fig. 67(a), the corresponding trajectory in the (ξ, η, ζ)-space tends to a circle (see Appendix A for more details). Initial condition: V (0) = 0.1,M(0) = 0.1,N(0) = 0.1. The initial point (V (0),M(0),N(0)) = (0.1, 0.1, 0.1) for Eq. (223) is projected into the point (ξ(0),η(0),ζ(0)) = (10.1, 0, 0.02) (10.1, 0, 0.141) via the coordinate transformation (229). (b) A chaotic trajectory of Eq. (223) under the sinusoidal forcing j(t) = 21sin(0.05t). Since Eq. (223) exhibits a chaotic oscillation in the (V,n, m,h)-space as shown in Fig. 67(b), the corresponding trajectory in the (ξ,η,ζ)-space is also chaotic (see Appendix A for more details). We deleted the transient trajectory to view the chaotic attractor clearly. Initial condition: V (0) = 0.1,M(0) = 0.1,N(0) = 0.1. (c) A chaotic trajectory of Eq. (223) under the rectangular-pulse forcing j(t) = 15.4 s[sin(0.05t)]. Since Eq. (223) exhibits a chaotic oscillation in the (V,n, m,h)-space as shown in Fig. 67(c), the corresponding trajectory in the (ξ,η,ζ)-space is also chaotic (see Appendix A for more details). We deleted the transient trajectory to view the chaotic attractor clearly. Initial condition: V (0) = 0.1,M(0) = 0.1,N(0) =

69 Chaotic Oscillation via Edge of Chaos Criteria (a) j(t) = 21sin(0.05t) (b) j(t) = 15.4 s[sin(0.05t)] Fig. 69. Poincaré maps of the forced Morris Lecar model defined by Eq. (223) with I = We can observe some folding loci. Furthermore, the chaotic attractors seem to be tightly compressed. The chaotic attractor in Fig. 69(a) looks like a one-dimensional curve. The chaotic attractor in Fig. 69(b) is split into several segments. We set ω 0 =0.05, which satisfies Re[Y (iω 0 )] < 0. We plotted the points (V (nt ),M(nT ),N(nT )) (n =1, 2,...)inthe(V,m,n)-space, where T = 2π = 2π ω denotes the period of the forcing signal. (a) Poincaré map under the sinusoidal forcing j(t) = 21sin(0.05t). Initial condition: V (0) = 0.1,M(0) = 0.1,N(0) = 0.1. (b) Poincaré map under the rectangular-pulse forcing j(t) = 15.4 s[sin(0.05t)]. Initial condition: V (0) = 0.1,M(0) = 0.1,N(0) = 0.1. [Sah et al., 2016]). We show the trajectories of Eq. (223) in Fig. 71. Here, the periodic forcing signals are given respectively by the sinusoidal function j(t) =20sin(0.2t), (231) and the rectangular-pulse function j(t) = 24.1 s[sin(0.2t)]. (232) In order to view their trajectories from a different perspective, let us project the trajectories into the (ξ,η,ζ)-space by the coordinate transformation: ξ(t) =(V (t) + 20) cos(0.2t), η(t) =(V (t) + 20) sin(0.2t), ζ(t) = M(t) 2 + N(t) 2. (233) (a) j(t) = 21sin(0.05t) (b) j(t) = 15.4 s[sin(0.05t)] Fig. 70. Average power P (t) of the forced Morris Lecar model defined by Eq. (223) with I = We set ω 0 =0.05, which satisfies Re[Y (iω 0 )] < 0. (a) P (t) < 0fort 0. Thus the average power delivered from the forced signal is not dissipated in the one-port connected across j(t). Here, the periodic forcing is given by j(t) = 21sin(0.05t). Initial condition: V (0) = 0.1,M(0) = 0.1,N(0) = 0.1. (b) P (t) > 0 for t > 0. Thus the average power delivered from the forced signal is dissipated in the one-port circuit. Here, the periodic forcing is given by j(t) = 15.4 s[sin(0.05t)]. Initial condition: V (0) = 0.1,M(0) = 0.1,N(0) =

70 M. Itoh & L. Chua (a) j(t) =0 (b)j(t) =20sin(0.2t) (c) j(t) = 24.1 s[sin(0.2t)] Fig. 71. Trajectories of the forced Morris Lecar model defined by Eq. (223) with I = We set ω 0 =0.2, which satisfies Re[Y (iω 0 )] > 0. (a) A trajectory tends to an asymptotically-stable equilibrium point as t if j(t) =0.Thecoordinate of the asymptotically-stable equilibrium point is given by (V 0,M 0,N 0 ) (7.799, , ). In this case, Eq. (223) is an autonomous system. Initial condition: V (0) = 0.1,m(0) = 0.1,n(0) = 0.1. (b) A chaotic attractor of Eq. (223) under the sinusoidal forcing j(t) = 20sin(0.2t). Initial condition: V (0) = 0.1,m(0) = 0.1,n(0) = 0.1. (c) A chaotic attractor of Eq. (223) under the rectangular-pulse forcing j(t) = 24.1 s[sin(0.2t)]. Initial condition: V (0) = 0.1,m(0) = 0.1,n(0) =

71 Chaotic Oscillation via Edge of Chaos Criteria (a) j(t) =0 (b)j(t) =20sin(0.2t) (c) j(t) = 24.1 s[sin(0.2t)] Fig. 72. Trajectories of Eq. (223) with I = , which are projected into the (ξ,η,ζ)-space via the coordinate transformation (233). We set ω 0 =0.2, which satisfies Re[Y (iω 0 )] > 0. Compare the trajectories in Fig. 72 with those in Fig. 68. Observe that the two chaotic attractors in Fig. 72 do not seem to be tightly compressed. That is, the orbits in Figs. 72(b) and 72(c) are more spread out (less dense) than those in Figs. 68(b) and 68(c). The trajectory is coded using the rainbow color code for ζ, that is, the color evolves through violet, indigo, blue, green, yellow, orange and red, as ζ varies from its minimum to its maximum value. (a) A trajectory for j(t) = 0, which converges to a thick orange circle after the transient regime. Since the trajectory in the (V,M,N)-space tends to an asymptotically-stable equilibrium point as shown in Fig. 71(a), the corresponding trajectory in the (ξ, η, ζ)-space tends to a circle (see Appendix A for more details). Initial condition: V (0) = 0.1,m(0) = 0.1,n(0) = 0.1. The initial point (V (0),M(0),N(0)) = (0.1, 0.1, 0.1) for Eq. (223) is projected into the point (ξ(0),η(0),ζ(0)) = (10.1, 0, 0.02) (10.1, 0, 0.141) via the coordinate transformation (233). (b) A chaotic trajectory under the sinusoidal forcing j(t) = 20[sin(0.2t)]. Since Eq. (223) exhibits a chaotic oscillation in the (V,n, m,h)-space as shown in Fig. 71(b), the corresponding trajectory in the (ξ,η, ζ)-space is also chaotic (see Appendix A for more details). We deleted the transient trajectory to view the chaotic attractor clearly. Initial condition: V (0) = 0.1,m(0) = 0.1,n(0) = 0.1. (c) A chaotic trajectory under the rectangular-pulse forcing j(t) = 24.1 s[sin(0.2t)]. Since Eq. (223) exhibits a chaotic oscillation in the (V,n, m,h)-space as shown in Fig. 71(c), the corresponding trajectory in the (ξ,η,ζ)-space is also chaotic (see Appendix A for more details). We deleted the transient trajectory to view the chaotic attractor clearly. Initial condition: V (0) = 0.1,m(0) = 0.1,n(0) =

72 M. Itoh & L. Chua (a) j(t) = 20sin(0.2t) (b) j(t) = 24.1 s[sin(0.2t)] Fig. 73. Poincaré maps of the forced Morris Lecar model defined by Eq. (223) with I = We set ω 0 =0.2, which satisfies Re[Y (iω 0 )] > 0. We plotted the points (V (kt),m(kt),n(kt)) (k =1, 2,...)inthe(V,M,N)-space, where T = 2π = 2π ω denotes the period of the forcing signal. We can observe many folding loci. However, these folds do not seem to be compressed tightly, compared with those in Fig. 69. (a) Poincaré map under the sinusoidal forcing j(t) = 20sin(0.2t). Initial condition: V (0) = 0.1,M(0) = 0.1,N(0) = 0.1. (b) Poincaré map under the rectangular-pulse forcing j(t) = 24.1 s[sin(0.2t)]. Initial condition: V (0) = 0.1,M(0) = 0.1,N(0) = 0.1. The projected trajectories are shown in Fig. 72. We also show the corresponding Poincaré maps in Fig. 73. We plotted the points (V (nt ),M(nT ), N(nT )) (n =1, 2,...)inthe(V,M,N)-space, where T = 2π = 2π denotes the period of the forcing ω signal. Note that Eq. (223) can exhibit a chaotic oscillation, even if Re[Y (iω 0 )] > 0. Compare the attractors in Fig. 73 with those in Fig. 69. Many folding loci can be observed in Fig. 73. However, these folds do not seem to be compressed tightly, compared with those in Fig. 69. We conclude as follows: Suppose that the angular frequency ω 0 of periodic forcing satisfies the edge of chaos condition (ii), that is, Re[Y (iω 0 )] < 0. Then, the chaotic attractor of the Morris Lecar model (220) with I = is tightly compressed. 7. Chaos Generation and Complexity The properties of the oscillators studied in this paper are summarized in Table 1. Observe that the Duffing oscillator does not satisfy the edge of chaos condition (ii), and the forced Van der Pol oscillator does not satisfy the edge of chaos condition (i). However, the remaining four oscillators can satisfy both edge of chaos conditions (i) and (ii). We can apply the following chaos generation procedure to the forced oscillators, which satisfy the edge of chaos criteria: Chaos generation procedure (i) Check that at least one equilibrium point is locally asymptotically stable when the system is unforced (i.e. autonomous). That is, all poles of Z(s) at the asymptoticallystable equilibrium point are on the open left-half plane. (ii) Choose a locally-active frequency ω 0, which satisfies Re[Z(iω 0 )] < 0. (iii) Apply a periodic forcing signal j(t) = A sin (ω 0 t). (iv) Tune the amplitude A so that a periodic response evolves to a chaotic oscillation. (v) If an amplitude A cannot be tuned to generate chaotic oscillation, then return to step (ii) with a different ω

73 Chaotic Oscillation via Edge of Chaos Criteria Table 1. Edge of chaos criteria. Stability of Equilibrium Sign of Re[Z(iω)] Edge of Chaos Edge of Chaos Oscillators Point Without Forcing or Re[Y (iω)] Condition (i) Condition (ii) Duffing oscillator Asymptotically stable Re[Z(iω)] > 0 for all ω > 0 Generalized Duffing oscillator Asymptotically stable Re[Z(iω 0 )] < 0forsomeω 0 Forced FitzHugh Nagumo model Asymptotically stable Re[Z(iω 0 )] < 0forsomeω 0 Forced Hodgkin Huxley model Asymptotically stable Re[Y (iω 0 )] < 0forsomeω 0 Forced Morris Lecar model Asymptotically stable Re[Y (iω 0 )] < 0forsomeω 0 Forced Van der Pol oscillator Unstable Re[Z(iω)] < 0 for all ω Since the chaotic oscillation takes place in narrow ranges of the circuit parameters, we might not be able to find a chaotic oscillation. In this case, we must return to step (ii) from step (v), and try to find a chaotic oscillation under a different set of circuit parameters. Furthermore, from the computer simulations in Secs. 2.11, 4.1 and 6.1, we conclude as follows: Phase transition The edge of chaos criteria can provide a phase transition from ordered to chaotic behavior by tuning the amplitude of the periodic forcing. Complex behavior Suppose an unforced system satisfies the edge of chaos conditions (i) and (ii). Then, the system can exhibit the following complex chaotic behaviors by periodic forcing. many folding loci, strong spiral loci, tight compressing loci. We summarize the relationship between global stability and the average power in Table 2. From this table, we can conclude as follows: Classification of forced oscillators The forced oscillators can be classified into two groups. In group 1, the oscillator dissipates average power, delivered from the sinusoidal current source. For example, Duffing oscillator, Forced Van der Pol oscillator. In group 2, the oscillators do not dissipate average power, but instead they generate energy via a locally-active circuit element inside the onepot N. For example, Generalized Duffing oscillator, Forced FitzHugh Nagumo model with I = and I =1.43, Hodgkin Huxley model with I = 9.7, Forced Morris Lecar model with I = The current source in this case, acts as a sink. Table 2. Global stability and average power P (t). Globally-Stable Equilibrium Point P (t) < 0 Oscillators Without Forcing? Under Sinusoidal Forcing? Duffing oscillator Generalized Duffing oscillator Forced FitzHugh Nagumo model with I = 1.43 Forced FitzHugh Nagumo model with I = Forced Hodgkin Huxley model with I = 9.7 Forced Morris Lecar model with I = Forced Van der Pol oscillator

74 M. Itoh & L. Chua Global stability and energy generation The following three oscillators have a globally asymptotically-stable equilibrium point if they are unforced (i.e. j(t) = 0): Generalized Duffing oscillator with j(t) =0, Forced FitzHugh Nagumo model with I =1.43, j(t) =0, Forced Morris Lecar model with I = , j(t) =0. Furthermore, they do not dissipate average power from the current source. Instead, they generate energy via chaotic oscillations, powered by the presence of at least one locallyactive circuit element inside the one-port N connected across the current source. The current source in these examples act as a sink. Note that the unforced FitzHugh Nagumo model with I = is not globally stable, since it has a stable limit cycle and an unstable equilibrium point [see Fig. 45(a)]. Furthermore, from our computer simulations, we observed the following behavior: Globally stable autonomous systems The following three oscillators have a globallystable equilibrium point when they are unforced, that is, j(t) = 0. In these cases, the resulting autonomous systems do not exhibit a limit cycle oscillation. Duffing oscillator, Generalized Duffing oscillator, Forced Morris Lecar model with I = Conclusion We have shown that the generalized Duffing oscillator, the FitzHugh Nagumo model, the Hodgkin Huxley model, and the Morris Lecar model can exhibit a chaotic oscillation when driven by a periodic input signal, when the one-port N satisfies the edge of chaos criteria. We have proposed a chaosgeneration procedure via the edge of chaos criteria. Furthermore, we have shown that certain kinds of asymptotically-stable nonlinear dynamical systems do not dissipate average power from the current source, but acts as an energy source instead. Our computer simulations have suggested that forced oscillators can exhibit complex chaotic behavior if the corresponding unforced oscillators satisfy the edge of chaos criteria. Acknowledgment This paper is supported in part by AFOSR grant FA References Andronov, A. A., Vitt, A. A. & Khaikin, S. E. [1987] Theory of Oscillators (Dover, NY). Chua, L. O. [2005] Local activity is the origin of complexity, Int. J. Bifurcation and Chaos 15, Chua, L., Sbitnev, V. & Kim, H. [2012] Neurons are poised near the edge of chaos, Int. J. Bifurcation and Chaos 22, Chua, L. [2013] Memristor, Hodgkin Huxley, and edge of chaos, Nanotechnology 24, Guckenheimer, J. & Holmes, P. [1983] Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields (Springer-Verlag, NY). Guckenheimer, J. & Oliva, R. A. [2002] Chaos in the Hodgkin Huxley model, SIAM J. Appl. Dyn. Syst. 1, Hodgkin, A. L. & Huxley, A. F. [1952] A quantitative description of membrane current and its application to conduction and excitation in nerve, J. Physiol. 117, Itoh, M. & Murakami, H. [1994] Chaos and canards in the Van der Pol equation with periodic forcing, Int. J. Bifurcation and Chaos 4, Itoh, M. & Chua, L. O. [2007] Oscillations on the edge of chaos via dissipation and diffusion, Int. J. Bifurcation and Chaos 17, Kanamaru, T. [2007] Van der Pol oscillator, Scholar- pedia 2, Morris, C. & Lecar, H. [1981] Voltage oscillations in the barnacle giant muscle fiber, J. Biophys. Soc. 35, Sah, M. P., Kim, H., Eroglu, A. & Chua, L. [2016] Memristive model of the Barnacle giant muscle fibers, Int. J. Bifurcation and Chaos 26, Thompson, J. M. T. & Stewart, H. B. [1986] Nonlinear Dynamics and Chaos, Geometrical Methods for Engineers and Scientists (Wiley, Chichester). Van Valkenburg, M. E. [1964] Network Analysis (Prentice-Hall, Englewood Cliffs, NJ)

75 Chaotic Oscillation via Edge of Chaos Criteria Appendix A Coordinate Transformation via the Parametric Equations of a Torus In this Appendix, we show that a trajectory in the two-dimensional plane can be transformed into a trajectory in the three-dimensional space, by using the parametric equations of a torus. A torus is the product of two circles, that is, a blue circle and a green circle as shown in Fig. 74. The blue circle is rotated around a red center circle inside the torus. Theequationofatorusinthe(ξ,η,ζ)-space is given by (R ξ 2 + η 2 ) 2 + ζ 2 = r 2, (A.1) where R is the radius of the red center circle, and r is the radius of the blue circle. A torus can also be defined parametrically by ξ =(R + r cos(θ)) cos(ϕ), η =(R + r cos(θ)) sin(ϕ), ζ = r sin(θ), (A.2) where θ and ϕ are parameters which defined two full circles, where 0 θ 2π and 0 ϕ 2π. Ifr =0, then Eq. (A.2) reduces to the parametric equations of the red center circle inside the torus. If we fix θ at some value, then Eq. (A.2) defines a green circle on the outer surface of the torus. Conversely, if we fix ϕ at some value, then Eq. (A.2) defines a blue circular cross-section at the fixed value ϕ, in Fig. 75. Fig. 75. Parameters of a torus: R is the radius of the red center circle, r is the radius of the blue circle, θ and ϕ are parameters which define each point on a circle, via Eq. (A.2), and they satisfy 0 θ 2π and 0 ϕ 2π. (x, y) = (r cos(θ),rsin(θ)). In this example, R =5andr =2.For the sake of simplicity, only the first quadrant of the (x, y)- plane is shown at ϕ = 0 (printed in a light yellow background color). The (x, y)-plane moves along the red center circle in an anticlockwise direction with angular speed ω. In order to apply Eq. (A.2) to the forced oscillators, let us define the variables x = r cos(θ), y = r sin(θ), (see Fig. 75). Then Eq. (A.2) is recast into ξ =(R + x)cos(ωs), η =(R + x)sin(ωs), ζ = y, (A.3) (A.4) where ϕ = ωs, ω is an angular frequency, and s is a parameter. Assume that x and y move on the circle with the radius r, thatis,x and y are defined by x(s) = r cos(ω 2 s), y(s) = r sin(ω 2 s), (A.5) where ω 2 is an angular frequency and s is a parameter. Then Eq. (A.4) is recast into ξ =(R + r cos(ω 2 s)) cos(ωs), Fig. 74. A torus is the product of two circles, that is, a green circle and a blue circle. The blue circle is rotated around the red closed circular axis inside the torus η =(R + r cos(ω 2 s)) sin(ωs), ζ = r sin(ω 2 s). (A.6)

76 M. Itoh & L. Chua The orbit defined by Eq. (A.6) moves on the torus in the (ξ,η,ζ)-space. We next explain the above coordinate transformation with some specific examples. Let us first consider the case where (x, y) moves on the unit circle with ω 2 = 3, that is, its orbit is defined by (x(s),y(s)) = (cos(3s), sin(3s)), (A.7) where s is a parameter, as shown on the left part of Fig. 76(a). In this case, Eq. (A.4) is recast into ξ(s) = (5 + cos(3s)) cos(0.2s), η(s) = (5 + cos(3s)) sin(0.2s), ζ(s) =sin(3s), (A.8) where we set R = 5 and ω = 0.2. Observe that Eq. (A.8) exhibits a periodic orbit in the (ξ,η,ζ)- space, as shown in Fig. 76(a). Consider next the case where the orbit on the (x, y)-plane is defined by (x(s),y(s)) = (cos( 7s), sin( 7s)), (A.9) (a) Periodic orbit in the (ξ,η,ζ)-space (b) Quasi-periodic orbit in the (ξ,η,ζ)-space Fig. 76. Relationship between the orbits on the (x, y)-plane and the (ξ,η,ζ)-space. (a) The orbit in the (ξ,η,ζ)-space is periodic when the orbit on the (x,y)-plane is given by (x(t),y(t)) = (cos(3t), sin(3t)). In this case, the angular frequency ratio for the periodic signals is given by 0.2 3, that is, it is a rational number. Here we set ϕ =0.2t and R = 5. (b) The orbit in the (ξ,η,ζ)-space is quasi-periodic when the orbit on the (x, y)-plane is given by (x(t),y(t)) = (cos( 7t), sin( 7t)). In this case, the angular frequency ratio for the periodic signals is given by 7 0.2, that is, it is not a rational number; namely, it is an irrational number. Here we set ϕ =0.2t and R = 5. (c) The orbit in the (ξ,η,ζ)-space converges to a red circle with a radius of 5 when the orbit on the (x, y)-plane is given by (x(t),y(t)) = (cos(3t)e 0.2t, sin(3t)e 0.2t ). Note that (x(t),y(t)) (0, 0) as t.herewesetϕ =1.5t, R =

77 Chaotic Oscillation via Edge of Chaos Criteria (c) Circular orbit (red circle) in the(ξ,η,ζ)-space Fig. 76. where s is a parameter. In this case, the angular frequency of x(s) andy(s) is not a rational number. The corresponding equation (A.4) is now given by ξ(s) =(5+cos( 7s)) cos(0.2s), η(s) =(5+cos( 7s)) sin(0.2s), ζ(s) =sin( 7s), (A.10) (Continued) where R = 5 and ω = 0.2. Equation (A.10) exhibits a quasi-periodic orbit in the (ξ,η,ζ)-space, as shown in Fig. 76(b). Consider next the case where the orbit on the (x, y)-plane converges to the origin along a shrinking spiral, as in a stable focus, thatis, (x(s),y(s)) = (cos(3s)e 0.2s, sin(3s)e 0.2s ), (A.11) (a) Chaotic orbit in the (ξ,η, ζ)-space Fig. 77. Relationship between the trajectories on the two-dimensional (x, y)-plane and on the three-dimensional (ξ, η, ζ)- space. The trajectory on the (x, y)-plane is transformed into the (ξ,η,ζ)-space via the coordinate transformation (A.14). The trajectory in the (ξ, η, ζ)-space is coded using the rainbow color code for ζ, that is, the color evolves through violet, indigo, blue, green, yellow, orange and red, as ζ varies from its minimum to its maximum value. (a) A trajectory in the (ξ, η,ζ)-space is chaotic when the trajectory on the (x, y)-plane is chaotic. We can view the chaotic attractor from a different perspective since the trajectory is transformed into a path in the three-dimensional (ξ,η,ζ)-space. Initial condition of Eq. (5): x(0) = 0.5,y(0) = 0.5. Parameters: R =5,s = t, ω =0.2, and A = 6. (b) A trajectory converges to a thick blue closed orbit in the (ξ,η,ζ)-space when the trajectory on the (x, y)-plane converges to a periodic orbit after the transient regime. Initial condition of Eq. (5): x(0) = 0,y(0) = 0. Parameters: R =5,s = t, ω =0.2, and A =0.5. (c) A trajectory converges to a thick green circle on the plane ζ = 0, with a radius of 5 in the (ξ,η,ζ)-space, after the transient regime when the trajectory on the (x, y)-plane converges to an equilibrium point (origin). Initial condition of Eq. (5): x(0) = 0.5,y(0) = 0.5. Parameters: R = 5, s = t, ω =0.2, and A =

78 M. Itoh & L. Chua (b) Periodic orbit (blue) in the (ξ,η,ζ)-space (c) Circular orbit (dark green circle) in the (ξ,η,ζ)-space Fig. 77. where s is a parameter. Then, Eq. (A.4) becomes ξ(s) = (5 + cos(3s)e 0.2s )cos(1.5s), η(s) = (5 + cos(3s)e 0.2s )sin(1.5s), ζ(s) =sin(3s)e 0.2s, (A.12) where R =5andω =1.5. In this case, the orbit in the (ξ,η,ζ)-space converges to a circle with a radius of 5 as shown in Fig. 76(c). Let us now apply the coordinate transformation (A.4) to the generalized Duffing oscillator defined by ẋ = y +0.02x, ẏ = 0.05y x 3 0.2x + j(t), (A.13) where j(t) = A sin(0.2t). Consider first the case where Eq. (A.13) exhibits chaotic behavior on the (x, y)-plane, as shown on the left side of Fig. 77(a). (Continued) Substituting the chaotic solution (x(t),y(t)) of Eq. (A.13) into Eq. (A.4), we obtain ξ(t) =(x(t)+5)cos(0.2t), η(t) =(x(t)+5)sin(0.2t), ζ(t) =y(t), (A.14) where we set R =5,s = t, ω =0.2, and A =6. In this case, a chaotic trajectory is not transformed into the orbit on the torus defined by Eq. (A.1). However, we can view the chaotic attractor from a different perspective as shown on the right side of Fig. 77(a). That is, the chaotic attractor on the (x, y)-plane is transformed into a chaotic attractor in the three-dimensional (ξ,η,ζ)-space. Consider next the case where Eq. (A.13) exhibits a periodic response on the (x, y)-plane after thetransientregime,asshownontheleftsideof Fig. 77(b). Substituting the solution (x(t),y(t)) of