Fractal-Like Basin Boundaries and Indeterminate Transition in Nonlinear Resonance of a Toda Oscillator

Abstract

We have numerically investigated the unpredictable jump in a nonlinear damped Toda oscillator. It was shown that three stable steady-states exist in a certain frequency region of the external force in this system, and these multiple steady-states are realized due to the overlap of the primary resonance branch with the secondary resonance branch. Even though no homoclinic tangles exist in the present system, we found that the unpredictable jump can occur via a saddle-node (SN) bifurcation, which occurs in the course of heteroclinic events. It is pointed out that the origin of the unpredictable jump (or unpredictable transition) is due to the complex basin structure which remains after the SN bifurcation.