Solitons, spatiotemporal chaos and synchronization in arrays of damped driven nonlinear oscillators

We explore periodic and chaotic motion in a soliton-bearing chain of nonlinear oscillators and study the effect of disorder on the spatiotemporal chaos. The chain we consider consists of torsionally coupled, damped, parametrically driven pendulums. We show that the amplitudes of the pendulums satisfy a system of equations in slow time, the “nonlinear Schrödinger (NLS) oscillators”. The evolution of the chain of NLS oscillators is simulated numerically. Simulating the homogeneous chain, we demonstrate the existence of periodically oscillating solitons, the period-doubling sequence of transitions to temporal chaos and the degeneration into spatiotemporal chaos for larger driving strengths. Next, we explore the effect of introducing weak disorder in the simplest form, a single impurity, into the chain. We describe how a long impurity can induce a pinned soliton to form and prevent spatiotemporal chaos, synchronizing the oscillators in the chain. Lastly, we study chains with several impurities in various configurations. Cases considered include chains with multiple impurities of different strengths, equal equidistant impurities (both long and short), and equal impurities which are positioned randomly along the chain. We show that all the previously reported features of the continuous NLS equation are reproduced in the dynamics of the discrete chain. Our results indicate that the dominant structures of the spatiotemporal chaos are unstable solitons. We show how spatiotemporally chaotic motion can be suppressed by disordering the chain through the inclusion of one or more impurities, synchronizing the oscillators. A comprehensive analysis of chains with identical equidistant long impui rities reveals an unexpectedly complex relationship between the number and strength of impurities and the dynamics observed. As a result of disordering the positions of impurities, we find that oscillating solitons form between impurities when the gaps are large enough. The oscillations may be periodic or chaotic. Equidistant short impurities may also stabilize the chain. We conclude that for a set of impurities to prevent spatiotemporal chaos from emerging in the array, the intervals between the impurities should be sufficiently small, and the strength of the impurities should be sufficiently large. The required number and strength of impurities depends on the oscillator parameters and the initial condition of the chain.