Global Bifurcations in a Damped-Driven Diatomic Granular Crystal

TL;DR

This paper investigates the complex dynamics of a damped-driven diatomic granular crystal, revealing how bifurcations and unstable manifolds lead to chaos, enhancing understanding of nonlinear phenomena in granular systems.

Contribution

It identifies unstable manifolds of saddle solutions and demonstrates their role in chaos emergence, providing new insights into bifurcation structures in granular crystals.

Findings

Unstable manifolds of saddle points are key to chaos onset.

Homoclinic tangles lead to chaotic attractors.

Period-doubling bifurcations destroy quasiperiodic tori.

Abstract

We revisit here the dynamics of an engineered dimer granular crystal under an external periodic drive in the presence of dissipation. Earlier findings included a saddle-node bifurcation, whose terminal point initiated the observation of chaos; the system was found to exhibit bistability and potential quasiperiodicity. We now complement these findings by the identification of unstable manifolds of saddle periodic solutions (saddle points of the stroboscopic map) within the system dynamics. We unravel how homoclinic tangles of these manifolds lead to the appearance of a chaotic attractor, upon the apparent period-doubling bifurcations that destroy invariant tori associated with quasiperiodicity. These findings complement the earlier ones, offering more concrete insights into the emergence of chaos within this high-dimensional, experimentally accessible system.