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

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. Copyright c © EPLA, 2013 Introduction. – Granular systems, consisting of densely packed particles that interact through nonlinear, tensionless potentials, have been recognized in the last decade as a fertile testbed where ideas from nonlinear dynamics can be put to use [1,2]. Relevant investigations have focused on the dynamics of nonlinear waveforms, including traveling waves [1–8] and discrete breathers [9], as well as other nonlinear processes, including second harmonic generation and nonlinear resonances [10]. Several potential applications have been suggested, including energy absorbing layers [11–13], sound scramblers [14], acoustic lenses [15], and rectifiers [16]. In the present work, we explore the ability of granular crystals to support chaotic dynamics, which are accompanied by energy transmission, when subjected to external driving that is above a critical amplitude and has a frequency in the forbidden gap of its linear spectrum. Dissipation is abundant in the system and has been the subject of recent investigations [17–19]. By driving the system, we produce a case example of a dampeddriven system of coupled nonlinear oscillators. This system has the potential for the future study of pattern formation [20,21] as well as nonlinear supratransmission [22,23], which can be triggered by the interplay of ordered (periodic, quasiperiodic) and chaotic dynamics [24]. Specifically, we study a one-dimensional, statically compressed diatomic system, consisting of alternating aluminum and steel spheres, where the first sphere is aluminum. In the case where the crystal starts with a steel sphere, its undamped and undriven Hamiltonian analogue does not support nonlinear surface modes, which leads to a drastic modification of the bifurcation features and dynamics detailed below. In our case, the presence of driving at the boundary and its interaction with dissipation, nonlinearity, and discreteness, enables two classes of relevant states: nonlinear surface modes and states tuned to the external driver. These two branches of time-periodic solutions are observed to collide and disappear in a limit cycle saddle-node bifurcation. Beyond