The double scroll family

Absrrucf-This paper provides a rigorous mathematical proof that the double scroll is indeed chaotic. Our approach is to derive a linearly equitdent class of piecewise-linear differential equations which includes the double scroll as a special case. A necessary and sufficient condition for two such piecewise-linear vector fields to be linearly equivalent is that their respective eigenvalues be a scaled version of each other. In the special case where they are identical, we have exact equivalence in the sense of hear conjugacy. An explicit normalform equation in the context of global bifurcation is derived and parametrized by their eigenvalues. Analytical expressions for various Poincur~ maps are then derived and used to characterize the birtli and the &4t/1 of the double scroll, as well as to derive an approximate one-dimensional map in analytic form which is useful for further bifurca-tion analysis. In particular, the analytical expressions characterizing various La/f-return maps associated with the Poincare map are used in a crucial way to prove the existence of a Shilnihov-type homoclinic orbit, thereby establishing rigorously the chaotic nature of the double scroll. These analytical expressions are also fundamental in our in-depth analysis of the birth (onset of the double scroll) and &&I (extinction of chaos) of the double scroll. The unifying theme throughout this paper is to analyze the double scroll system as an unfolding of a large family of piecewise-linear vector fields in R3. Using this approach, we were able to prove that the chaotic dyrumics of the double scroll is quite common, and is robust because the associated horseshoes predicted from Shilnikov's theorem are structurally stable. In fact, it is exhibited by a large family (in fact, infinitely many linearly-equideni circuits) of vector fields whose associated piecewise-linear differential equations bear no resemblance to each other. It is therefore remarkable that the normalized eigenvalues, which is a local concept, completely determine the system's global qualitative behavior.