Homoclinic Intersections and the Macroscopic Observability of Arnold Tongues in the Forced-Dissipative Duffing System

Abstract

Dissipative chaos often exhibits abrupt transitions to periodic windows or vanishing states, but these transitions occur far below macroscopic theoretical boundaries such as the Melnikov threshold. In this study, we re-evaluate the transversal intersections (microscopic) of invariant manifolds from a deterministic and entropic perspective for the ``Arnold tongues'' shown by synchronization to forced inputs in the parameter space. We applied an algorithm that digitally determines manifold intersections as binary values ($1.0$ or $0.0$) without numerical interpolation. As a result of scanning the $Ω- F$ parameter plane at a resolution of $5000 \times 5000$ ($25$ million points) with the damping coefficient fixed at $k = 0.2$, it became possible to globally capture the relationship between the macroscopic phase-locked regions shown by the Arnold tongues and the microscopic manifold intersections (chaotic regions). Furthermore, from a one-dimensional cross-section whose computational accuracy was verified, we confirmed a dynamical case where the region with homoclinic intersections (topological entropy $h_T > 0$) is a necessary condition for the region where chaos manifests (Kolmogorov-Sinai entropy $h_{KS} > 0$), and simultaneously confirmed the existence of a region where the intersection of the primary saddle solution does not serve as a necessary condition.