Generalizations of the Chua equations

We present two generalizations of the equations governing Chua´s circuit. In the type-I generalization of Chua´s equations we use a 2-D autonomous flow as a component in a 3-D autonomous flow in such a way that the resulting equations will have double-scroll attractors similar to those observed experimentally in Chua´s circuit. The value of this generalization is that (1) it provides a building block approach to the construction of chaotic circuits from simpler 2-D components that are not chaotic by themselves. In so doing, it provides an insight into how chaotic systems can be built up from simple nonchaotic parts; (2) it illustrates a precise relationship between 3-D flows and 1-D maps. In the type-II generalized Chua equations we show that attractors similar to the Lorenz and Rossler attractors can be produced in a building block approach using only piecewise linear vector fields. As a result we have a method of producing the Lorenz and Rossler dynamics in a circuit without the use of multipliers. These results suggest that the generalized Chua equations are in some sense fundamental in that the dynamics of the three most important autonomous 3-D differential equations producing chaos are seen as variations of a single class of equations whose nonlinearities are generalizations of the Chua diode