Chaotic Systems and Circuits with Hidden Attractors

Transcription

1 haotic Sstems and ircuits with Hidden Attractors Sajad Jafari, Viet-Thanh Pham, J.. Sprott Abstract ategoriing dnamical sstems into sstems with hidden attractors and sstems with selfecited attractors is new topic in dnamical sstems. In this chapter we describe three newl introduced families of chaotic sstems with hidden attractors. We design a circuit for one eample of each famil and discuss some important properties of these kinds of circuits.. Introduction Recent research has involved categoriing periodic and chaotic attractors as either self-ecited or hidden [-]. A self-ecited attractor has a basin of attraction that is associated with an unstable equilibrium, whereas a hidden attractor has a basin of attraction that does not intersect with small neighborhoods of an equilibrium points. The classic attractors of Loren, Rössler, hua, hen, Sprott (cases B to S), and other widel-known attractors are self-ecited with one or more unstable equilibrium points. From a computational standpoint, this allows one to use a numerical method in which a trajector started from a point on the unstable manifold in the neighborhood of the unstable equilibrium, reaches an attractor and identifies it [7]. Hidden attractors cannot be found b this method and are important in engineering applications because the allow unepected and potentiall disastrous responses to perturbations in a structure like a bridge or an airplane wing. It has been shown in [-3] that the chaotic attractors in dnamical sstems without an equilibrium points, with onl stable equilibria, or with a line of equilibria are eamples of hidden attractors. That ma be the reason wh such sstems are rarel found, and onl a few such eamples have been reported in the literature [-3]. These sstems are challenging, and studing them ma reveal new phenomena in dnamical sstems. On the other hand, the circuit implementation of chaotic sstems has attracted much interest in the past decades [4-3]. It has bolstered the confidence that chaos is a real phenomenon and has provided eperimental data for man applications in the stud of chaos. In the net section, we briefl introduce the above mentioned three families of hidden attractors. In the third section we show a designed circuit for one eample of each famil. We conclude with some discussion in the last section where we mention some important properties of chaotic circuits with hidden attractors.. Three new families of hidden chaotic attractors

2 .. haotic flows with no equilibria In this section we consider chaotic flows with no equilibria. Such sstems have neither homoclinic nor heteroclinic orbits [3], and thus the Shilnikov method [3, 33] cannot be used to verif the chaos. The oldest and best-known eample is the conservative Sprott A sstem [34] listed as NE in Table. This is an important sstem since it is a special case of the Nose-Hoover oscillator [35] which describes man natural phenomena [36], and thus it suggests that such sstems ma have practical as well as theoretical importance. In [] we performed a sstematic search to find three-dimensional chaotic sstems with quadratic nonlinearities and no equilibria. Our search was based on the methods proposed in [37] and used our own custom software. The objective was to find the algebraicall simplest cases which cannot be further reduced b the removal of terms without destroing the chaos. The main method for finding these sstems was to constrain the search space to cases where we could show algebraicall that the equilibrium points are imaginar. For eample, an chaotic solution of a parametric sstem such = = = k + k + k 3 + k 4 + k 5 + k 6 + k 7 + k 8 + k 9 + a () k 4k a 4 is a candidate. An ehaustive computer search was done considering man thousands of combinations of the coefficients and initial conditions, seeking cases for which the largest Lapunov eponent is greater than.. In addition to the seventeen cases listed in the Table, doens of additional cases were found that are etensions of these cases with additional terms. For each case that was found, the space of coefficients was searched for values that are deemed elegant [37], b which we mean that as man coefficients as possible are set to ero with the others set to ± if possible or otherwise to a small integer or decimal fraction with the fewest possible digits. Ecept for NE, all these cases are dissipative. The Lapunov spectra and Kaplan-Yorke dimension are shown in Table along with initial conditions that are close to the attractor. Table Seventeen simple chaotic sstems with no equilibria. Model Equations a LEs D KY (,, ) NE =..38,, (, 5, )

3 = = a = NE = ,, (,.4, ) = + a = NE 3 =..5,, (,, -) = a =. + a NE 4 = +..35,, (-8.,, -5) = 3 = NE 5 =..68,, (.98,.8, -.7) = + + a = NE 6 =.75.8,, (, 3, -.) NE 7 = a = = + = a..5,, (,.3, ) = NE 8 =.3.34,, (,., )

4 = +.5 a = NE 9 =.55.54,, (.5,, ) = + 7 a = NE =.6.6,, (,.7,.8) NE = a = = + =.8 a..76,, (,.6, 3) = NE =..654,, (.5,, -) = a = NE 3 = +.4.8,, (.5,, ) = + +. a = NE 4 =..53,, (,, -4) = a = NE 5 =..,, (,, -4.9) = a

5 NE 6 = a = a = a..67,, (,, -) = a NE 7 = a..57,, (, -, ) = a.. haotic flows with stable equilibria In [], the search for chaotic flows with a stable equilibrium first focused on jerk sstems. onsider a general equation with quadratic nonlinearities of the form = = (,, ) = f () f = a + a + a + a + a + a + a + a + a + a An equilibrium point of ( *, *, * ) of sstem () must have * = * = and eigenvalues that satisf λ f λ f λ f 3 (3) in which f = a + a 4 *, f = a + a 7 *, and f = a 3 + a 8 *. Using the Routh-Hurwit stabilit criterion, we require f <, f f + f >, and f < for that equilibrium to be stable. We can find * from a + a 4 * + a =. For a 4, we have = ( ± ) a / a, 4 where = a 4a a. To have an equilibrium, should be greater than or equal to ero, in which 4 * * case, = ( a + ) / a and 4 ( ) a / a =. From the stabilit condition f < for *, we 4 have <, which is impossible. Thus a quadratic jerk sstem cannot have two stable equilibria, and we therefore modif the general case in Eq. (3) to

6 = = (4) = a + a + a + a + a + a + a + a + a in which there is no term in the equation to ensure that one and onl one equilibrium eists. This sstem has a single equilibrium at ( a / a,,) whose stabilit requires a < 9 a a a < a (5) a a a a < 6 9 a a7a9 a3 a Again, an ehaustive computer search was done considering man thousands of combinations of the coefficients a through a 9 and initial conditions subject to the constraints in Eq. (5), seeking cases for which the largest Lapunov eponent is greater than.. ases SE -SE 6 in Table are si simple cases found in this wa. As can be seen, the eigenvalues for the equilibria at the origin have all negative real parts, which means that each equilibrium is stable. B similar calculations, man other simple structures for chaotic flows were investigated, and 7 additional cases (SE 7 -SE 3 ) were added to the previous si jerk sstems in Table. In addition to the cases listed in the table, doens of additional cases were found, but the were either equivalent to one of the cases listed b some linear transformation of variables, or the were etensions of these cases with additional terms. The Lapunov spectra and Kaplan-Yorke dimensions are shown in Table along with initial conditions that are close to the attractor. Since all the cases have a stable equilibrium, a point attractor coeists with a strange attractor for each case. Table. Twent-three simple chaotic flows with one stable equilibrium. Model Equations Equilibrium Eigenvalues LEs D KY (,, ) SE = = = ±.749i

7 SE = = = ±.9896i SE 3 = = = ±.959i SE 4 = = = ±.7683i SE 5 = = = ±.4i SE 6 = = = ±.9744i ±.968i ±.9987i ±.997i

8 .5.5 ±.9997i 5.7 /..47 ±.999i / ±.i / ±.9948i ±.84i / ±.93i ±.57i /3..57/3...9 ±.767i ±.4483i

9 ±.8497i ±3.5875i ±.879i ±.4483i ±.6i haotic flows with line equilibria In [3], the search for chaotic flows with a line equilibrium was inspired b the structure of the conservative Sprott case A sstem [34], (6) We consider a general parametric form of Eq. (6) (without the constant term in ) with quadratic nonlinearities of the form (7)

10 As can be seen, this sstem has a line equilibrium at (,, ) with no other equilibria (in other words the -ais is an infinite line equilibrium of equilibrium). As before, an ehaustive computer search was done considering millions of combinations of the coefficients a through a 9 and initial conditions, seeking dissipative cases for which the largest Lapunov eponent is greater than.. ases LE -LE 6 in Table 3 are si simple cases found in this wa with onl si terms. With a similar procedure, three other similar cases LE 7 -LE 9 were found and included in Table 3. The equilibria, eigenvalues, Lapunov eponent spectra, and Kaplan-Yorke dimensions are shown in Table 3 along with initial conditions that are close to the attractor. A discussion of wh these sstems belong to the categor of hidden attractors is in [3]. Table 3 Si simple chaotic flows with a line equilibrium. ase Equations (a,b) Equilibrium Eigenvalues LEs D KY (,, ) LE a = 5 b = LE a = 7 b = LE 3 a = 8 b = LE 4 a = 4 b = LE 5 a =.5 b =

11 LE 6 a =.4 b = LE 7 a =.85 b = LE 8 a = 3 b = LE 9 a =.6 b = ircuit realiation of three new families of hidden chaotic attractors 3.. ircuit realiation of one chaotic flow with no equilibria The electronic chaos generator of the chaotic flow with no equilibrium NE 6 was designed (Fig. ). The circuit consists of common electronic components such as resistors, operational amplifiers, capacitors and multipliers. The variables of the sstems,, correspond to the voltages of capacitors,, and 3, respectivel. Hence the dnamics of the proposed circuit can be epressed as three differential equations in the corresponding voltage variables v, v, v 3 as dv R8 = v, dt R R7 dv R = v, 3 dt R R9 dv 3 = v v. v v 3 v V 3 a dt R33 R43 R53 R63 (8)

12 The values of the components are chosen as follows: R = R = R3 = R6 = R7 = R8 = R9 = R = kω, R4 = R5 = kω, = = 3 = nf, and Va.75V =.The designed circuit was simulated using the D electronic simulation package Multisim for the initial conditions v ( ), v ( ), v ( ) = V, 3 V,.V. The resulting phase portraits shown in Fig. are ( ) ( ) similar to the theoretical ones in []. Fig.. ircuit realiation of the chaotic sstem with no equilibrium NE 6.

13 a b c Fig.. Simulation results of the designed circuit for the NE 6 sstem using Multisim software: a) portrait b) v v phase portrait c) v 3 v phase portrait 3 v v phase

14 3.. ircuit realiation of one chaotic flow with stable equilibria An analog circuit emulating the chaotic flow SE with a stable equilibrium is illustrated in Fig. 3. The schematic consists of eleven resistors (from R to R ), three capacitors (,, and 3 ), and five operational amplifiers (from U to U 5 ). B appling Kirchhoff s law to the circuit in Fig. 3, the dnamics of the designed circuit is described b the circuit equations dv R9 = v, dt R R8 dv R = v, 3 dt R R dv 3 = + dt R R R R R v v v v 3 v 3 v (9) where, v v, v, 3 are the voltages across the capacitors,, and 3, respectivel. The values of the electronic components in Fig. 3 are selected to match known parameters of the chaotic flow SE with a stable equilibrium: R = R = R3 = 6 kω, R 5 = 3 kω, R 6 =.6 kω, R 7 =.5 kω, R 4 = R 8 = R 9 = R = R = kω, = = 3 = nf. The proposed circuit in Fig. 3 was simulated using the electronic simulation package Multisim with initial conditions of v ( ), v ( ), v ( ) = 4 V, V, V. The resulting chaotic attractors are shown in Fig. 4. ( ) ( )

15 Fig. 3. The proposed circuit which emulates the chaotic flow with a stable equilibrium SE.

16 a b c Fig. 4. haotic attractors ehibited b the circuit in Fig. 3: a) v v v phase portrait b) v phase portrait 3 v v phase portrait c) 3

17 3.3. ircuit realiation of one chaotic flow with a line equilibria An analog circuit was also designed to realie the chaotic flow with a line equilibrium LE. The schematic of the proposed circuit is shown in Fig. 5. The circuit equations, which are derived b appling Kirchhoff s law to the circuit in Fig. 5, can be written as dv R8 = v, dt R R7 dv = v + v, v 3 dt R R3 dv dt R R R 3 = v v v v v () Four operational amplifiers, eight resistors, three capacitors, and three multipliers were used. The values of the components are R = R = R4 = R7 = R8 = kω, R 3 = R 6 = kω, R 5 =.666 kω, and = = 3 = nf. Multisim simulations were implemented with the initial ( v, v, v ) ( V,.5 V,.5V ) conditions ( ) ( ) ( ) =. The resulting phase portraits are shown in Fig. 6. Fig. 5. Schematic of the designed analog circuit for the chaotic sstem with a line equilibrium LE.

18 a b c Fig. 6. Simulation results of the designed circuit for the LE sstem using Multisim software: a) v portrait b) v v phase portrait c) v 3 v phase portrait 3 v phase

19 3. 4. Discussion There are some precautions for designing an analog circuit that emulates the dnamics of a chaotic flow, especiall when the attractor is hidden: a) The components of the analog circuit must be selected carefull to match the mathematical model. hoosing correct off-the-shelf discrete components is a practical challenge. For eample, it is eas to change the parameters and the eigenvalues of the chaotic sstem with one stable equilibrium when inappropriate values of the electronic components are used. In such cases, the dnamics of the sstem can change radicall. b) The limits of operational amplifiers and analog multipliers, such as saturation, power suppl voltages, nonlinearities, frequenc limitations, and acceptable inputs must be considered. For eample in a chaotic sstem like NE 4, the amplitudes of the variables are much greater than the other cases (see Fig. in []). c) A special sub-circuit generating the initial conditions for the voltages on the capacitors should be implemented to provide the appropriate initial conditions. Setting appropriate initial conditions is essential for realiing a chaotic flow with coeisting attractors since such a circuit will not oscillate if the initial conditions are in the basin of the stable equilibrium. In fact, this is the main difference between chaotic circuits with hidden attractors and those with self-ecited attractors. Since the attractor is hidden, it will not be observed when using initial conditions close to an equilibrium. d) B the abovementioned discussion, it seems that the chaotic flows with one stable equilibrium should not be suggested as student projects in the limit time of a course, since the are more difficult to construct (as we eperienced). 4. onclusion ategoriing dnamical sstems into sstems with hidden attractors and sstems with selfecited attractors is a new topic in dnamical sstems. We described three newl introduced families of chaotic sstems with hidden attractors (chaotic attractors in dnamical sstems without an equilibrium points, with onl stable equilibria, and with a line of equilibria). We designed a circuit for one eample of each famil. We described the difficulties in implementing circuits with hidden attractors resulting from the necessit of carefull choosing component values and initial conditions.

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