An approach toward higher dimensional hysteresis chaos generators

An approach toward higher dimensional autonomous chaotic circuits is discussed. Special consideration is given to a particular class of circuits which includes only one nonlinear element, namely, a three-segment piecewise-linear resistor, and one small inductor, L ₀, serially connected with it. A simple four-dimensional example that realizes hyperchaos is given. For the case in which L ₀ is shorted, the circuit equation can be simplified to a constrained system and a two-dimensional Poincare map can be rigorously derived. This mapping and its Lyapunov exponents verify laboratory measurements of hyperchaos and related phenomena. A rigorous approach to the singular perturbation theory of an N-dimensional circuit family that includes the above example is then provided. A canonical equation which describes any circuit in this family is derived. This equation can also be simplified to a constrained system, and an (Nn-2)-dimensional Poincare map can be derived. The main theorem indicates that this mapping explains the observable solutions of the canonical form very well