GENERALIZING THE TWIST-AND-FLIP PARADIGM

Abstract

In this paper we generalize the horseshoe twist theorem of Brown and Chua [1991] and derive a wide class of ODEs, with and without dissipation terms, for which the Poincare map can be expressed in closed form as FTFT where T is a generalized twist. We show how to approximate the Poincaré maps of nonlinear ODEs with continuous periodic forcing by Poincare maps which have a closed-form expression of the form FT 1 T 2 … T n where the T i are twists. We extend the twist-and-flip map to three dimensions with and without damping. Further, we demonstrate how to use the square-wave analysis to argue for the existence of a twist-and-flip paradigm for the Poincare map of the van der Pol equation with square-wave forcing. We apply this analysis to the cavitation bubble oscillator that appears in Parlitz et al. [1991] and prove a variation of the horseshoe twist theorem for the twist-and-shift map, which models the cavitation bubble oscillator. We present illustrations of the diversity of the dynamics that can be found in the generalized twist-and-flip map, and we use a pair of twist maps to provide a specific and very simple illustration of the Smale horseshoe. Finally, we use the twist-and-shift map of the cavitation bubble oscillators to demonstrate that the addition of sufficient linear damping to a dynamical system having PBS (Poincaré–Birkhoff–Smale) chaos may cause the chaos to become detectable in computer simulations.