Hysteresis loops and multi-stability: From periodic orbits to chaotic dynamics (and back) in diatomic granular crystals

¹ Department of Mathematics and Statistics, University of Massachusetts - Amherst, MA 01003-4515, USA
² Graduate Aerospace Laboratories (GALCIT) California Institute of Technology - Pasadena, CA 91125, USA
³ Department of Mechanical Engineering, Massachusetts Institute of Technology - Cambridge, MA 02139, USA
⁴ Department of Chemical and Biological Engineering and PACM, Princeton University Princeton, NJ, 08544, USA

Received: 2 November 2012
Accepted: 20 January 2013

Abstract

We consider a statically compressed diatomic granular crystal, consisting of alternating aluminum and steel spheres. The combination of dissipation, driving of the boundary, and intrinsic nonlinearity leads to complex dynamics. Through both numerical simulations and experiments, we find that the interplay of nonlinear surface modes with modes caused 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, we observe hysteretic dynamics and an interval of multi-stability involving stable periodic solutions and chaotic ones.