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EXPERIMENTAL POINCARE MAPS FROM THE

TWIST-AND-FLIP CIRCUIT

by

Guo-Qun Zhong, Leon O. Chua, and Ray Brown

Memorandum No. UCB/ERL M95/110

1 October 1995

EXPERIMENTAL POINCARE MAPS FROM THE

TWIST-AND-FLIP CIRCUIT

by

Guo-Qun Zhong, Leon O. Chua, and Ray Brown

Memorandum No. UCB/ERL M95/110

1 October 1995

ELECTRONICS RESEARCH LABORATORY

College of EngineeringUniversity of California, Berkeley

94720

Experimental Poincare Maps from the Twist-and-Flip Circuit

Guo-Qun Zhong* Leon 0. Chua, and Ray Brown

Abstract

In this paper, we present a physical implementation of the twist-and-flip circuit containinga nonlinear gyrator. Many phase portraits and their associated Poincari maps are observedexperimentally from this circuit and presented in this paper.

1. Introduction

Fractals are one of many manifestations of complicated chaotic dynamics. The fractal phenomenon can occur not only in autonomous systems typical of Chua's circuit [l]-[3], but alsoin nonautonomous systems driven by time-varying signals and therefore described by a nonau-tonomous system of ordinary differential equations

x = /(x,t) (1)

where x is a vector in an n-dimensional Euclidean space Rn.The twist-and-flip circuit offers one ofthesimplest paradigms for nonautonomous chaos. Indeed,

the state equations associated with the twist-and-flip circuit are the only known nonautonomoussystem of ordinary differential equations whose Poincare map can be derived in an explicit analyticform. Based on this property ofthe circuit an in-depth mathematical analysis ofthe twist-and-flipmap has been carried out exhaustively and rigorously [4]-[5]. The various fractals correspondingto several classes of nonlinear gyration conductance functions g(vi,v2) from this map have beengenerated numerically [6].

In this paper we describe a physical implementation of the twist-and-flip circuit with a simplenonlinear gyration conductance function g(vu v2), driven by a square-wave voltage source. A variety of phase portraits and the corresponding Poincare maps observed experimentally from thissetup will be presented.

2. Physical implementation of the twist-and-flip circuit

The twist-and-flip circuit contains simply two linear capacitors C\ and C2, a voltage source s(t),and a nonlinear gyrator, as shown in Fig. 1(a). The voltage source s(t) for driving the circuit is asquare wave of amplitude a and angular frequency w(or period P = 2e), as shown in Fig. 1(b).The gyrator, which is the only nonlinear element in thecircuit, is described by the equations

•He ison leave from Guangzhou Institute ofElectronic Technology, Academia Sinica, Guangzhou 510070, People'sRepublic of China.

ii=9(vuV2)v2 1 (2)«2 = -0to,V2)vi J

where $(vi,v2) is the associated gyration conductance [7]. In this paper we assume that

g(vuv2) > 0, for -oo < t>i,v2 < oo.

2.1 State equations of the twist-and-flip circuit

Applying KCL and KVL to the circuit shown in Fig. 1(a), we obtain

ti =-Cjf \ (3)i2 = -C2|f/

where

v1(t) = x(t)-s(t),v2(t) = y(t).

Introducing equations (2) into equations (3) and normalizing C\ = C2 = 1, we obtain the following state equations governing the dynamics ofthe twist-and-flip circuit:

%= -g(x-s(t),y)y (4a)%= g(x-s(t),y)(x-s(t)) (4b)

where

s(t) = a, t 6 (n, n + \)P (positive-half cycle)s(t) = -a, t € (n+ |,n+ 1)P (negative-half cycle)

n = 0,l,2,---.

Dividing (4b) by (4a), we obtain

g =-<2=4 (5a)and

ili _ _i*±«l (5m

over each positive- and negative-half cycles, respectively. Obviously, equation (5) defines a phaseportrait consisting of a family of concentric circles, centered at x =• a overeach positive-half cyclete(nyn+ \)P, and at x= -o over each negative-half cycle t€{n+\,n+ 1)P, n= 0,1,2, -•*, asshown in Figs. 2(a)-(b) . The phase portrait over a full period is shown in Fig. 2(c).

2.2 Implementation of the twist-and-flip circuit

To implement the twist-and-flip circuit shown in Fig. 1(a), we must specify the nonlinear gyration conductance function g(v\,v2) for the circuit. In this paper we choose

p(t;i,v2) = 0>i+t>2. (6)

The block diagram of the physical implementation for the twist-and-flip circuit is shown inFig. 3(a). It consists of two squaring functional blocks, two multipliers, two voltage-controlledcurrent sources, one adder, one square-root functional block, two linear capacitors C\ and C2, anda square-wave voltage source s(t). The two squaringfunctional blocks, the adder, and the square-root functional block together perform the task to realize the gyration conductance function (6).The two analog multipliers and voltage-controlled current sources together realize the equation (2).The capacitors C\ and C2 are connected across port 1 and port 2 of the gyrator, respectively. Theinput signal to the twist-and-flip circuit is s(t), a square-wave voltage source with amplitude a andangular frequency w. The analog multipliers AD633JN and AD734AN are respectively used toimplement all multiplications, squaring, and square-rooting operations in Fig. 3(a). The completecircuit diagram of the twist-and-flip circuit is shown in Fig. 3(b).

3. Experimental observations of Poincare maps in the twist-and-flip circuit

Though the twist-and-flip circuit has an explicit mathematical solution, its asymptotic behavior in the x - y phase plane is nevertheless muddled by an infinite tangle of intersections of thetrajectory upon itself. The standard method to untangle such a mess ofpoints and extract someuseful asymptotic information is to analyze the dynamics of the associated Poincare map definedas follows: Given any point (so,ifo), the Poincare map of (x0,3fo) for the circuit is a point (zi,yi)that corresponds to the position of the trajectory (originating from (x0yyo)) at t = P= -jj [6].

Fig. 4shows the circuit setup used in our experimental observations of the Poincare map. Thesampled-and-hold output signals x and yof the circuit are connected respectively to the horizontaland vertical channels ofan oscilloscope, operating in X-Y mode. The sample-and-hold impulse isgenerated by the square-wave voltage source s(t) used for driving the twist-and-flip circuit.

By changing the amplitude a or the period P of the square-wave voltage source «(/), a richvariety of phase portraits, including chaotic and periodic behaviors, have been observed from theimplemented twist-and-flip circuit. The experimentally constructed phase portrait, and its corresponding Poincare map in the x - y plane, are shown in Figs. 5(a)-(f). Note that the phaseportrait in the x- y plane is muddled by an infinite tangle of intersections of the trajectory uponitself, whereas the corresponding Poincare map in the x- y plane is a clearly defined set of pointsillustrating the chaotic attractor occurring in the twist-and-flip circuit. To compare with theoreticresults, the Poincare maps obtained by numerical simulation for the circuit operating in a chaoticregime are shown in Figs. 6(a)-(f). The simulation is based on Eqs. (4a) and (4b), in which capacitances Ci and C2 are normalized to 1. The twist-and-flip circuit is theoretically lossless. However,the physical realization of this circuit must necessarily involve some losses, however small. Theselosses, or damping factors are nonlinear and very diflicult tomodel exactly. Therefore, toconstructa reasonable simulation requires some approximations tothese losses. The way to include dampingin the twist-and-flip mapping is suggested by the dissipative example in [4], and the derivations ofdissipative Poincare maps in [5]. Using these results we include two dissipative parameters in thesimulations, a, and /?. The resulting simulations of the Poincare maps shown in Figs. 6(a)-(f) havethe same voltages and frequencies utilized in the circuit corresponding to Figs. 5(a)-(f). Whilethe correspondences are not exact, the differences we believe are due tothe difficulties of capturingthe exact form ofthe nonlinear dissipative terms to be included in the twist-and-flip maps used tosimulate the circuit. Given this qualification, several ofthe Poincare" maps occurring in Fig. 5 arein excellent geometric agreement to what is seen in typical dissipative twist-and-flip maps. From

this we conclude that this circuit is a good electronic realization of the twist-and-flip paradigm ofchaos.

The trajectory is stable when the circuit operates in a periodic regime. In this case, the phaseportrait in the x - y plane is stationary, and the Poincare map is a set ofperiodic points. When asquare wave ofamplitude a = 0 over each negative-half cycle is used as the sample-and-hold voltage,we obtain the display shown in Figs. 7(a)-(b), in which the phase portrait and its Poincare mapover each positive-half cycle of the square-wave voltage source s(t) appear simultaneously.

4. Concluding remarks

The most remarkable property of the twist-and-flip circuit is that its associated nonautonomousstate equations have an explicit and simple Poincare map, making it mathematically tractable. Thiscircuit is imbued with a full repertoire of complicated chaotic dynamics. In this paper , we haveimplemented the twist-and-flip circuit having a nonlinear gyration conductance functiondefined byp(t>i, v2) = Jv\ + v2, and presented simulated numerically observations utilizing the twist-and-flipmap model with losses. Due to the complex nature of the actual model for the losses and thesensitivity of the dissipative Poincare map to the nonlinear dissipative parameters, constructingthe exact damping model is difficult and is an open question for further study. Utilizing the samecircuit synthesis technique, a rich variety of Poincare maps can be observed experimentally bysynthesizing other twist-and-flip circuits having more complicated gyration conductance functions,such as those presented in [6].

Acknowledgment

This work was supported in part by the Office of Naval Research under Grant N00014-89-J-1402and by the National Science Foundation under Grant MIP 86-14000.

One of the authors, G. Q. Zhong, would like to thank C. W. Wu for helpful discussions.

References

[1] L. O. Chua, "The genesis of Chua's circuit," Archiv Elektronik Ubertragungstechnik, vol. 46,no. 4, pp. 250-257,1992.

[2] R. N. Madan, Guest Ed., Special Issue on "Chua'sCircuit: A Paradigmfor Chaos", J. Circuits,Syst. Computers, Part I: Mar. 1993; Part II: June 1993.

[3] G. Q. Zhong and F. Ayrom, "Experimental confirmation of chaosfrom Chua's circuit," Int. J.Circuit Theory Appl. vol. 13, no.l, pp. 93-98, 1985.

[4] R. Brown and L. O. Chua, "Horseshoes in the twist-and-flip map," Int. J. Bifurcation Chaos,vol. 1, no. 1, pp. 235-252, 1991.

[5] R. Brown and L. O. Chua, "Generalizing the twist-and-flip paradigm ," Int. J. BifurcationChaos, vol. 1, no. 2, pp. 385-416,1991.

[6] L. 0. Chua, R. Brown, and N. Hamilton, "Fractals in the twist-and-flip circuit," Proc. ofIEEE, vol. 81, no. 10, pp. 1466-1491, Oct. 1993.

[7] L. O. Chua, Introduction to Nonlinear Network Theory. McGraw-Hill, 1969.

Figure Captions

Fig. 1 (a) Twist-and-flip circuit containing two hnear capacitors and a gyrator characterized bya nonlinear gyration conductance function g(vi, t>2), driven by a voltage source s(t).(b) Voltage source s(t) in (a), a square wave with amplitude a and angular frequency a>.

Fig. 2 (a) Phase portrait centered at x = a over each positive-half cycle of s(t).(b) Phase portrait centered at x = —a over each negative-half cycle of s(t).(c) Combined phase portrait over a full period of s(t).

Fig. 3 (a) Block diagram of physical implementation for the twist-and-flip circuit,(b) Implemented circuit diagram of the twist-and-flip circuit.

Fig. 4 Experimental setup for observing the Poincare map.

Fig. 5 Experimentally observed phase portrait (left) and its corresponding Poincare map (right)of the twist-and-flip circuit, driven by a square-wave voltage source s(t) with the circuitoperating in a chaotic regime. Horizontal axis is x, vertical axis is y. Parameter values: (a)a = 1.0V, P = 1.8ms; (b) a = 1.5V, P = 2.4ms; (c) a = 2.0V, P = 1.35ms; (d) o = 2.0V, P =1.7ms; (e) a = 2.5V, P = 1.3ms; (f) a = 3.0V, P = 0.81ms.

Fig. 6 Numerically simulated Poincare map for the circuit operating in a chaotic regime. Horizontal axis is 3, vertical axis is y. Parameter values: (a) a = 1.0V,P = 1.8ms,a = 0.013,0 = 3;(b) a= 1.5V, P= 2.4ms, a = 0.026, 0= 2.3 ; (c) a= 2.0V, P= 1.35ms, a = 0.02,0 = 1.5; (d)a = 2.0V,P = 1.7ms,a = 0.02,0 = 2.5; (e) a = 2.5V,P = 1.3ms,a = 0.020605,0 = 2.2686.(f) a = 3.0V, P = 0.81ms, a = 0.006,0 = 5.8.

Fig. 7 Phase portrait and Poincare map experimentally observed with the circuit operating in achaotic regime. Horizontal axis is x, vertical axis is y. a = 3.0V, P = 2.4ms.(a) Phase portrait in x-y plane; (b) Corresponding Poincare map superimposed on the phaseportrait over each positive-half cycle ofs(t).

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