Chaos after Accumulation of Torus Doublings

Abstract

A recent paper investigates the bifurcation diagrams involved with torus doubling and asserts that the chaotic attractors observed after torus doubling have two Lyapunov exponents that are exactly zero. Against this assertion, we claim that the absolute value of one of the calculated zero Lyapunov exponents is not exactly zero but is instead slightly positive, because successive torus doubling is constrained by a very small underlying parameter. We justify our position by calculating Lyapunov spectra precisely using an autonomous piecewise-linear dynamical circuit. Our numerical results show that one of the Lyapunov exponents is close to, but not exactly, zero. In addition, we consider coupled logistic and sine-circle maps whose dynamics express the fundamental mechanism that causes torus doubling, and we confirm that torus doubling occurs fewer times when the coupling parameter of this discrete dynamical system is relatively larger. Consequently, the absolute value of the second Lyapunov exponent of this discrete system does not approach zero after the accumulation of torus doubling when the coupling parameter is set to larger values.