Theoretical Analysis For Chaotic Circuits

Since 1990, chaotic communications and chaotic encryption have become active research fields. As a fundament of these applications, studies on chaotic circuits, including the existence and mechanism of chaotic phenomena, the stability of the performance of chaotic circuit and so on, are all important problems faced to us.This paper summarizes the way to analysis chaotic circuits with computer and proposes some results on several typical chaotic circuits. The existence of a horseshoe is looked as the essential character of a chaotic system. It not only can be used to proof the existence of chaos, but also can be used to find out the mechanism of chaos and design chaotic system. So the topological horseshoe theory is the main tool. In this paper, we analysis a 2D Wien-bridge based chaotic circuit, find a 1D Poincarémap on an corresponding invariable set which semi-conjugates a 4-shift map, and then prove that the circuit is indeed chaotic. By similar way, we study the 3D Chua's circuit and its 2D Poincarémap, and verify the existence of chaos in the circuits. Since hyperchaos can exhibit much more complex dynamics than common chaos with only one positive Lyapunov exponent, it is believed that hyperchaos can play a better role in most applications of chaos. However the high demission and complexity make the studies of hyperchaos too hard. So, we study horseshoes of two significant hyperchaotic circuits. By numerical computation, we find horseshoes with two dimensional expansions in both of the circuits. In this way, we verify the existence of hyperchaos, and illustrate the mechanism of the hyperchaotic phenomena. Since the topological theory can only used to study chaotic system with one group of parameters, we also study Lyapunov exponents, bifurcation plots of these circuits.