Chaos in the Duffing oscillator beyond the Melnikov criterion

In this paper, we revisit the classical perturbed Duffing system and investigate its intricate dynamical behavior through the numerical method based on the topological horseshoe theory employing the Runge-Kutta method. Based on the classical Melnikov analysis, we explore the persistence of chaotic dynamics beyond the parameter regimes in which the Melnikov condition guarantees the existence of a transverse homoclinic intersection. Specifically, we examine the second return map and demonstrate the existence of a topological horseshoe at parameter values εγ=0.4, εδ=0.54, and ω=1. This provides numerical evidence to Smale horseshoe-type chaos in a regime where the Melnikov criterion is not satisfied. Furthermore, we provide a more rigorous treatment on the existence of the topological horseshoe in terms of crossing stability.