Amplitude-phase control of a novel chaotic attractor

Transcription

1 Turkish Journal of Electrical Engineering & Computer Sciences journals. tubitak. gov. tr/ elektrik/ Research Article Turk J Elec Eng & Comp Sci (216) 24: 1 11 c TÜBİTAK doi:1.396/elk Amplitude-phase control of a novel chaotic attractor Chunbiao LI 1,2,3,, İhsan PEHLİVAN 2,4, Julien Clinton SPROTT 2 1 School of Electronic & Information Engineering, Nanjing Universit of Information Science & Technolog, Nanjing, P.R. China 2 Department of Phsics, Universit of Wisconsin-Madison, Madison, WI, USA 3 Engineering Technolog Research and Development Center of Jiangsu Circulation Modernization Sensor Network, Jiangsu Institute of Commerce, Nanjing, P.R. China 4 Department of Electrical and Electronics Engineering, Facult of Technolog, Sakara Universit, Esentepe Campus, Serdivan, Sakara, Turke Received: Accepted/Published Online: Final Version: Abstract: A novel chaotic attractor with a fractal wing structure is proposed and analzed in terms of its basic dnamical properties. The most interesting feature of this sstem is that it has complex dnamical behavior, especiall coexisting attractors for particular ranges of the parameters, including two coexisting periodic or strange attractors that can coexist with a third strange attractor. Amplitude and phase control methods are described since the are convenient for circuit design and chaotic signal applications. An appropriatel chosen parameter in a particular quadratic coefficient can realize partial amplitude control. An added linear term can change the smmetr and provide an accessible knob to control the phase polarit. Finall, an amplitude-phase controllable circuit is designed using PSpice, and it shows good agreement with the theoretical analsis. Ke words: Fractal structure, amplitude control, phase control, coexisting attractors 1. Introduction Chaotic sstems and their dnamical behavior have inspired the interest of researchers because of their potential application in man fields, especiall in the communication and information industries. Chaotic attractors are a signature of chaos and reveal its character. Recentl, strange attractors with wing structures [1 11], such as the four-wing attractor [2 7] and multiwing attractor [8 11], have been of interest because of their elegant smmetr and multiple possible extensions. Furthermore, a chaotic flow generall has complexit and diversit. Some nonlinear dnamical sstems exhibit hsteresis and coexisting attractors [12,13]. Wing structures not onl consist of the main attractor branches, but also provide a bond between different parts of the phase trajector. In this paper, we propose a novel chaotic attractor different from other ones [14 16], which exhibits four winglike branches with a bond that has a double-wing structure. In this sense, the wing structure can penetrate the different regions of the attractor and the transition bonds. For selected parameters, two and even three coexisting attractors are observed. On the other hand, adapting a chaotic sstem to engineering applications often requires snchronization [17 19] and amplitude-phase control [2,21]. However, the amplitude and the phase control parameters have not been discussed in chaotic sstems with quadratic nonlinearit. In this paper, to achieve adequate amplitude Correspondence: chunbiaolee@gmail.com 1

2 and phase control, we introduce a parameter in a quadratic coefficient to achieve partial amplitude control and to change the smmetr of the chaotic sstem to provide phase polarit control. A corresponding chaotic circuit is designed in PSpice, which confirms the theoretical analsis. 2. Novel chaotic attractor Liu and Chen proposed a relativel simple three-dimensional continuous-time autonomous chaotic sstem [22] as follows. ẋ = ax z, ẏ = b + xz, ż = cz + x, This sstem possesses multiple smmetries in the equations and dnamical flow. Linear and nonlinear terms can be added to change the smmetr and behavior. We introduce a linear term b in the third dimension of Eq. (1) to control the dnamics. ẋ = x z, ẏ = a + xz, ż = b cz + x, When a = 2.5, b = 3.75, and c = 1.125, sstem (2) displas a novel chaotic attractor with a wing structure as shown in Figure 1 with initial conditions of (x,, z ) = (, 1, 1). The equations are solved with MATLAB using a fourth-order Runge Kutta integrator with a fixed time step of.5. The added linear term is necessar to produce a real four-wing chaotic sstem. The calculated Lapunov exponents are L 1 =.2712, L 2 =, and L 3 = , and the Kaplan Yorke dimension is D KY = 2 L 1 /L 3 = Figure 2 shows cross-sections of the attractor in the planes x = 3 and = 3. These sections show six main branches, two of which consist of subbranches in what appears to be a fractal structure as expected for a strange attractor. (1) (2) 3. Basic dnamical properties 3.1. Equilibria Sstem (2) has five equilibria that be can be found from the following. x z =, a + xz =, b cz + x =, (3) For α = b 2, β = ab 2 +4a 2 c 2 a, µ = b 2 a, ω = ab 2 +4a 2 c 2a, the equilibria are P 1 = (,, ), P 2 = (α + β, µ ω, a), P 3 = (α + β, µ + ω, a), P 4 =( α β, µ + ω, a), and P 5 = (α β, µ ω, a). When a = 2.5, b = 3.75, and c = 1.125, the corresponding equilibrium points are P 1 = (,, ), P 2 = (4.396, , ), P 3 = (4.396, , ), P 4 = (.646,.451, ), and P 5 = (.646,.451, ). Equilibrium points P 2 and P 3, and P 4 and P 5 are smmetrical about the x-axis. 2

3 (a) z (c) z x 2 z 1 2 x 1 x 1 2 (d) x x Figure 1. Chaotic attractor: (a) three-dimensional view, x phase plane, (c) x z phase plane, (d) z phase plane. 2 2 Z Z 2 22 Y X 22 Figure 2. Cross-sections of the attractor in planes where (a) x = 3, = 3. 3

4 3.2. Jacobian matrices The Jacobian matrix for sstem (2) at P 1 = (,, ) is: J 1 = 1 z z a x b + x c = (4) From λi J 1 =, the eigenvalues of the Jacobian matrix J 1 are λ 1 = 1, λ 2 = 1.125, and λ 3 = 2.5. Since λ 1 is a positive real number and the other two are negative real numbers, the equilibrium P 1 is a saddle-node. For the equilibrium P 2, the Jacobian matrix is: J 2 = 1 z z a x b + x c = (5) Here the eigenvalues are λ 1 = , λ 2 = i, and λ 3 = i. λ 1 is a negative real number, and λ 2 and λ 3 are a pair of complex conjugate eigenvalues with positive real parts. Therefore, the equilibrium P 2 is a saddle-focus. For the third equilibrium P 3, the eigenvalues of J 3 are identical to J 2, and thus it is also a saddle-focus. For the equilibrium points P 4 and P 5, the eigenvalues are identical and given b λ 1 = , λ 2 = i, and λ 3 = i, which is also a saddle-focus. Thus, sstem (2) is unstable at all equilibrium points Smmetr characteristics Sstem (1) has a unique rotational smmetr with respect to each of the axes as evidenced b its invariance under the coordinate transformations (x,, z) (x,, z),(x,, z) ( x,, z), and(x,, z) ( x,, z), respectivel. An added linear term in an dimension can change the smmetr. The term b in sstem (2) selects a specific smmetr with respect to the x-axis since the equations are then invariant onl under the coordinate transformation (x,, z) (x,, z). All equilibrium points are also smmetric with respect to the x-axis. Thus, the added term changes the smmetr and provides an accessible knob to control the phase polarit. 4. Amplitude-phase control and coexisting attractors 4.1. Amplitude control Although it is possible to linearl rescale all the variables to avoid exceeding amplitude limitations of the hardware, it turns out that partial amplitude control is achieved b adding a coefficient d in the quadratic term z in the first dimension of sstem (2). ẋ = x dz, ẏ = a + xz, ż = b cz + x, If we take d kd, x x, / k, z z/ k(k > ) in Eq. (6), the resulting sstem is identical to sstem (2), which means that parameter d can control the amplitude of variables and z according to 1/ k while leaving the variable x unchanged. (6) 4

5 LI et al./turk J Elec Eng & Comp Sci The expected behavior is verified b numerical calculations as shown in Figure 3, which also shows that to within numerical error, the Lapunov exponent spectrum is independent of d, as desired. This works because the first equation of sstem (6) has a single nonlinear term, and the second two equations are first-order in and z, suggesting a general principle. Conversel, d would be a bad parameter to choose for studing bifurcations. (a) (c) (d) Figure 3. Amplitude parameter d adjusted the range of and z without altering x: (a) x d bifurcation diagram at =, z d bifurcation diagram at =, (c) range of variables, (d) Lapunov exponents Phase control and coexisting attractors Sstem (2) has three quadratic terms and four linear terms. Amplitude control parameters generall involve the coefficient of a nonlinear term, while phase control involves the coefficient of a linear term. In sstem (2), phase polarit can be controlled b changing the sign of b. The sstem is invariant to the transformation b b, x x,, z z, and thus b controls the phase of xand as shown in Figure 4. The bifurcation diagrams for x and are reverse smmetrical, and the Lapunov exponent spectrum is smmetrical in the negative and positive regions, confirming that changing the sign of b reverses the polarit of x and, independent of whether the dnamics are chaotic. In particular, when b is chosen at 3.75 and 5

6 LI et al./turk J Elec Eng & Comp Sci 3.75 with initial values (, 1, 1) and (, 1, 1), respectivel, the phases of the chaotic signals x(t) or (t) are reversed as shown in Figure 5 despite the sensitive dependence on initial conditions. Generall, when phase polarit is reversed, the signs of the corresponding initial conditions also need to be changed to remain within the basin of attraction Figure 4. Bifurcation dnamics at z = when b varies in [ 15, 15]: (a) x b bifurcation diagram, b bifurcation diagram, (c) Lapunov exponents (with initial values (, 1, 1) and (, 1, 1) in the negative and positive regions, respectivel). Furthermore, sstem (2) has complex dnamical behavior in the region [ 15, 15], including the existence of coexisting attractors. Some chaotic attractors are shown in Figure 6 for various values of b. For b =.5 and an initial condition of (, 1, ± 5), sstem (2) has two two-wing attractors as shown in Figure 6a, where the dashed line is for the initial condition (, 1, 5). Moreover, when b 1.37, sstem (2) has two attractors, which are periodic at b = With an increase in b, the two two-wing structures coalesce and become a single four-wing attractor as shown in Figures 6b and 6c. For b greater than about 6, sstem (2) has three attractors, two of which are smmetric in z. For b = 7.8, the chaotic attractor coexists with two stable limit ccles as shown in Figure 6d, where the limit ccles are shown with a dashed line for the initial conditions (7, ±13, ±23). When 6

7 (a) 2 1 b= 3.75 b= b= 3.75 b=3.75 x t t Figure 5. Phase polarit control: (a) x versus t, versus t. (a) (c) 2 15 (d) Figure 6. Phase portrait in the z plane with initial values (, 1, 5) for (a) b =.5, b = 1.6, (c) b = 2.8, and (d) b =

8 b = 9.5, sstem (2) has its maximum largest Lapunov exponent of L 1 =.968 and Kaplan Yorke dimension of D KY = Chaotic circuit design Amplitude control is important for circuit design and engineering applications because of hardware limitations. To avoid saturating analog multipliers and operational amplifiers, the circuit signal amplitude is usuall reduced b a linear rescaling of the variables. However, man chaotic sstems with smmetr, like sstem (2), include a knob to achieve partial amplitude control. In such a case, total amplitude control requires additional control onl for the remaining uncontrolled variable(s). In Figure 1, we see that all of the signals, x,, and z, oscillate in the interval of ( 2, 2). Considering the amplitude control of and z b the parameter d, onl the variable x needs to be rescaled. Letting u = x/2, v =, and w = z, and then setting the original state variables x,, z instead of the variables u, v, w, the rescaled phase-controllable sstem becomes the following. ẋ = x d 2 z, ẏ = a + 2xz, ż = ±b cz + 2x, (7) From Eq. (7), we design the amplitude-phase adjustable analog circuit shown in Figure 7. The circuit equations in terms of the circuit parameters are: ẋ = 1 R 1 C 1 x 1 R 2 C 1 z, ẏ = 1 R 3C R 4C 2 xz, ż = ± 1 R 5C 3 1 R 6C 3 z + 1 R 7C 3 x, (8) The circuit consists of three channels to realize the integration, addition, and subtraction of the state variables x,, and z, respectivel. The operational amplifier OPA44/BB and its peripheral circuit perform the addition, inversion, and integration, and the analog multiplier AD633/AD performs the nonlinear product operation. The state variables x,, and z in Eq. (8) correspond to the state voltages of the three channels, respectivel. For the sstem parameters a = 2.5, b = 3.75, c = 1.125, and d = 4, the circuit element values are R 1 = 4 kω, R 2 = R 4 = R 7 = 2 kω, R 3 = 16 kω, R 5 = 1.66 kω, R 6 = kω, R 7 = 2 kω, and R 8 = R 9 = R 1 = R 11 = 1 kω. We select the capacitor C 1 = C 2 = C 3 = 1 nf to obtain a stable phase portrait, which onl affects the time scale of the oscillation. Figure 8 shows the phase portraits predicted b PSpice circuit simulation. Unlike other chaotic circuits, here a potentiometer R 2 is required to realize amplitude control. Generall, we first put the resistance at an extreme value and then adjust the control knob so that the voltages satisf the requirements of the hardware. With a decrease in R 2, the amplitude of and z decrease accordingl. Furthermore, phase polarit control is implemented in the circuit. A polarit change of b is realized b the switch that provides the appropriate feedback of the signal. The corresponding phase trajectories predicted b PSpice simulation are shown in Figure 9. 8

9 Figure 7. Amplitude-phase adjustable chaotic circuit. (a) 5.V 5.V V V 5.V 5.V 5.V V 5.V V(Y) V(X) (c) 8.V 6.V 5.V V 5.V V(Z) V(X) 4.V 2.V V 2.V 4.V 6.V 8.V 8.V 4.V V 4.V V(Z) V(Y) Figure 8. Phase portraits of sstem (2) in PSpice simulation with initial values x() =, () = 1, andz () = 1: (a) attractor in the x plane (1 V/div), x z plane (1 V/div), and (c) z plane (1 V/div). 9

10 5.V 5.V V V 5.V 5.V 8.V 4.V V 4.V 8.V 4.V V 4.V V(Y) V(X) V(Z) V(X) (a) Figure 9. Phase polarit of x and reversed: (a) phase portrait in the x plane (1 V/div) and x z plane (1 V/div). 6. Conclusion An added linear term in a pseudo four-wing sstem extracts the main four-wing structure with a special wing bond and produces a novel chaotic attractor. Its dnamical behavior is analzed, especiall coexisting attractors and amplitude and phase control. Due to the smmetr and sstem structure, a control method for amplitude and phase is established b introducing a parameter in the coefficient of a specific nonlinear term to achieve partial amplitude control, and a parameter is added in a linear term to alter the smmetr and allow phase polarit control. The corresponding amplitude-phase adjustable circuit is designed. The amplitude of two signals can be smoothl controlled, and the size of the corresponding attractors can be adjusted b a potentiometer. Furthermore, the phase polarit can be controlled b a simple switch. The results of PSpice circuit simulation show good agreement with the theoretical analsis. These special properties of the sstem provide a convenient passage to realize amplitude adjusting and phase adjusting in electronic engineering, especiall in communication sstems and radar sstems. Even though the modified version seems more complicated and costl, the simple knobs to realize amplitude and phase control actuall save some extra circuits to construct a much more complicated sstem. Acknowledgments The research described in this publication was supported b the China Postdoctoral Science Foundation (Grant No. 211M5838, 212T5456) and the Postdoctoral Science Foundation of Jiangsu Province (Grant No. 124C), and also in part b the Jiangsu Overseas Research Training Program for Universit Prominent Young and Middle-Aged Teachers and Presidents, the 4th 333 High-Level Personnel Training Project (Su Talent [211] No. 15) of Jiangsu Province. References [1] Özoğuz S, Elwakil AS, Kenned MP. Experimental verification of the butterfl attractor in a modified Lorenz sstem. Int J Bifur Chaos 22; 12:

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