FROM THE CHUA CIRCUIT TO THE GENERALIZED CHUA MAP

Abstract

We analytically derive a one-dimensional map from an ODE which produces a double scroll very similar to the Chua double scroll. Our analysis leads us to suggest a generalization of the Chua circuit to an n-dimensional system of ODEs that we will call the generalized Chua equations. The third order system of ODEs in this class contains the Chua equations as a special case. Parallel to the generalized Chua equations we define the generalized Chua maps. An important feature of these equations and maps is that the source of their nonlinearity is a sigmoid function, and functions very similar in their properties to the sigmoid function. We show that this class of equations contains examples of maps that reproduce the Lorenz and Rössler dynamics. We suggest that a general theory of these equations and maps, and their relationship to one-dimensional maps, is possible. A benefit of our analysis shows that the dynamics of the maps of Rössler, Chua, and Lorenz maps can be traced to a common set of building blocks, and we conclude that the Chua map is the simplest of the three maps and therefore understanding the complexity in the Chua map provides a foundation for understanding chaos in a large class of n-dimensional equations that includes the maps of Rössler and Lorenz.