A Four-Dimensional Plus Hysteresis Chaos Generator

This paper discusses a four-dimensional plus hysteresis autonomous chaotic circuit. The circuit dynamics is described by two symmetric four-dimensional linear equations connected to each other by hysteresis switchings. We transform the equation into Jordan form and derive theoretical formulas of its three-dimensional return map, its Jacobian matrix and its Jacobian. These formulas can be developed easily to general dimensional cases and are used to evaluate Lyapunov exponents. Then we have discovered torus doubling route to chaos and then to hyperchaos. Some of the return map attractors are confirmed by laboratory experiments. A rough two parameters bifurcation diagram is also given. I. INTR~DUC~~N C HAOTIC phenomena in electric circuits have been studied with great interest. In the study of autonomous chaotic circuits, some interesting results are given for threedimensional (3-D) systems [l]-[5], and some experimental results are given for four-dimensional (4-D) ones [6]-[8]. Then more higher dimensional systems have been recently begun to investigate [9]. More than 3-D circuits can exhibit hyperchaos [6] and related interesting phenomena which cannot be observed in 3D ones. Hyperchaos is a higher dimensional chaos introduced by R&sler [7] and is usually defined as a chaotic attractor with more than one positive Lyapunov exponent. It implies that its dynamics expand more than one direction. They relate important fundamental problems: classification of chaos, route to chaos and so on [lo], [II]. Also analysis and synthesis of such circuits may contribute to engineering applications, among them: spread spectrum communications [ 121, [ 131, controlling chaos [14]-[ 161 and memory search in artificial neural networks [ 171. However, the analysis of more than 3-D chaotic circuits is difficult because of the system complexity. In order to approach to such a circuit, we should focus on a simple model. Then, this paper considers a 4-D plus hysteresis autonomous circuit given by Fig. l(a). Here, -r-i and -r-s are linear negative resistors characterized by V; = -rjij (j = 1,2). In experiments, we utilize the central part of a current-controlled nonlinear resistor characterized by Fig. l(b). Fig. l(c) gives its implementation example. -H is a dependent voltage source Manuscript received April 11, 1994; revised September 5, 1994. This paper was recommended by Associate Editor Michael Peter Kennedy. The authors are with the Department of Electrical and Electronical Engineering, HOSE1 University, Koganei-shi, Tokyo, 184 Japan. IEEE Log Number 9407 179. characterized by the following hysteresis (see Fig. l(d)): -H(v1 +v2) = E for VI + 212 2 -Era/r6 -E for VI + 712 < Era/q, ’ (1) -H is switched from E to -E if vi +va hits the left threshold -Er,/rb and vice versa. Fig. l(e) gives an implementation example of -H. Here, -H consists of inverting adder and hysteresis comparator. Hereafter, we assume that the op amp is linear and that the zener diode is ideal. The circuit dynamics can’be described by two symmetric linear equations connected to each other by a hysteresis switchings: RCg = Ril (q + H(vl + vz)),Ll$ = -VI + rlil RC% = Ri2 (~1 +H(q +vz)),Lz$f = -7~2 +r&. (2) Here the vector field of the state variables consists of two overlapping 4-D halfspaces. Such hysteresis chaos generators have been developed by Newcomb’s group and us. Newcomb and El-leithy have proposed a chaos generator that includes binary hysteresis [2] in 1984. Contemporarily, the second author has proposed a hysteresis chaos generator based on a quasi-harmonic oscillator [ 181. Also we show a chaotic circuit family that includes one hysteresis resistor in [19] and the normal form equation from five-dimensional case is equivalent to (7) in some parameter range. The hysteresis resistor can be realized by three segments piecewise linear resistor for which a small inductor La is connected in series. Letting La tend to zero, the piecewise linear resistor is to be hysteresis one. Then the 3and 4-D circuit can be treated as 2and 3-D plus hysteresis one, respectively. In these cases, we have given a sufficient condition for chaos generation under strong parameter restriction [5], [8]. Chua’s circuit includes a three segments piecewise linear resistor and some interesting results are given by using piecewise linear techniques [20], [21]. The 2-D plus hysteresis chaos generator [2], [5] is a limiting case of Chua’s circuit and its analysis procedure is simpler because of hysteresis switching of two linear systems. This circuit exhibits interesting phenomena. Fig. 2 shows some examples of them as rb/r, decreases. Fig. 2(a) shows a periodic orbit. This circuit has two different resonance frequencies controlled by two inductors L1 and L2, respectively, and their interaction affects the dynamics. As rb/r, decreases, the attractor changes to torus Fig. 2(b) and then to chaotic attractors Fig. 2(c) and (d). The chaos Fig. 2(d) has different 1057-7122/94$04.00