Spatio-temporal dynamics of coupled array of Murali-Lakshmanan-Chua circuits

Abstract

The circuit recently proposed by Murali, Lakshmanan and Chua (MLC) is one of the simplest nonautonomous nonlinear electronic circuits which show a variety of dynamical phenomena including various bifurcations, chaos and so on. In this paper we study the spatiotemporal dynamics in one- and two-dimensional arrays of coupled MLC circuits both in the absence as well as in the presence of external periodic forces. In the absence of any external force, the propagation phenomena of traveling wavefront and its failure have been observed from numerical simulations. We have shown that the propagation of the traveling wavefront is due to the loss of stability of the steady states (stationary front) via subcritical bifurcation coupled with the presence of neccessary basin of attraction of the steady states. We also study the effect of weak coupling on the propagation phenomenon in one-dimensional array which results in the blocking of wavefront due to the existence of a stationary front. Further we have observed the spontaneous formation of hexagonal patterns (with penta{hepta defects) due to Turing instability in the two-dimensional array. We show that a transition from hexagonal to rhombic structures occur by the influence of an external periodic force. We also show the transition from hexagons to rolls and hexagons to breathing (space-time periodic oscillations) motion in the presence of external periodic force. We further analyze the spatiotemporal chaotic dynamics of the coupled MLC circuits (in one dimension) under the influence of external periodic forcing. Here we note that the dynamics is critically dependent on the system size. Above a threshold size, a suppression of spatiotemporal chaos occurs, leading to a space-time regular (periodic) pattern eventhough the single MLC circuit itself shows a chaotic behavior. Below this critical size, however, a synchronization of spatiotemporal chaos is observed.