The overdamped double‐scroll family. Part I: Piecewise‐linear geometry and normal form

Abstract

AbstractThis paper is the first part of a two part work that aims to introduce a new family of piecewise‐linear differential equations that exhibit chaotic behaviour in numerical simulations and to provide a rigorous mathematical proof of its chaotic nature.The new family presented here has the same eigenvalue distribution as the well‐known Lorenz equations in its chaotic regime but has a different symmetry. It is a derivative of the recently investigated double‐scroll family of piecewise‐linear differential equations.1 It differs, however, from the double‐scroll family in that its members have all real eigenvalues at the origin. In particular, since the associated eigenspace Es(0) is similar to that of an overdamped linear circuit, we will henceforth refer to this family as the overdamped double‐scroll family.We establish Chua's circuit2 as a member of this family under appropriate conditions. Preliminary numerical simulations exhibit some interesting phenomena such as period‐doubling, periodic windows between chaotic behaviour, and strange attractors of differing geometric structure.Our approach in the rigorous analysis will be that employed for the double‐scroll family in Reference 1. We derive a linearly equivalent class of piecewise‐linear differential equations which includes the family simulated numerically as a special case. the necessary and sufficient condition for two piecewise‐linear vector fields belonging to this new overdamped double‐scroll family to be linearly equivalent is that their respective eigenvalues at respective equilibrium points be scalar multiples of each other. If the scalars are all identity, then we have linear conjugacy of the vector fields. an explicit normal form equation, in the sense of global bifurcation, is presented that is parametrized by its own eigenvalues.We find that linearly equivalent vector fields associated with the overdamped double‐scroll family exhibit the same global behaviour even though their equivalence is based on the local concept of normalized eigenvalues. In addition, the piecewise‐linear differential equations associated with these equivalent vector fields can be quite different from each other.