Hysteresis Loops and Multi-stability: From Periodic Orbits to Chaotic Dynamics (and Back) in Diatomic Granular Crystals

TL;DR

This study investigates complex nonlinear dynamics, including chaos and hysteresis, in a driven diatomic granular crystal, combining theoretical, computational, and experimental approaches to understand bifurcations and multi-stability.

Contribution

It provides the first detailed analysis of hysteresis, multi-stability, and chaos in a diatomic granular crystal driven at the boundary, supported by experiments.

Findings

Chaotic dynamics emerge beyond saddle-node bifurcations.

Hysteresis and multi-stability are observed between periodic and chaotic states.

Chaotic energy propagation is weakly dependent on driving frequency.

Abstract

In the present work we consider a diatomic granular crystal, consisting of alternating aluminum and steel spheres, where the first sphere is an aluminum one. The combination of dissipation, driving of the boundary, and intrinsic nonlinearity leads to complex dynamics. Specifically, we find that the interplay of nonlinear surface modes with modes created by the driver create the possibility, as the driving amplitude is increased, of limit cycle saddle-node bifurcations beyond which the dynamics of the system becomes chaotic. In this chaotic state, part of the applied energy can propagate through the chain. We also find that the chaotic branch depends weakly on the driving frequency and speculate a connection between the chaotic dynamics with the gap openings between the spheres. Finally, a reverse parametric continuation reveals hysteretic dynamics and the existence of an interval of multi-stability involving stable periodic solutions and genuinely chaotic ones. The computational identification and theoretical interpretation of the bifurcation diagram of the system is corroborated by direct experimental measurements showcasing all of the above features.