Dynamic behaviors of a symmetrically coupled period-doubling system

Abstract

A system of two coupled mappings demonstrates a variety of nonlinear phenomena such as the inverse state, spatiotemporal intermittence, traveling wave and the synchronization. In this paper, we are concerned with a system of symmetrically coupled quadratic mappings. Bezruchko et al. employed numerical method to study the bifurcation problem of such a system, but did not give a full investigation in theory because of the complicated computation. In this paper, we adopt the complete discrimination system theory and the real root isolation algorithm to overcome the difficulty. We will give a completed description of the bifurcations in theory for such a system, including the transcritical bifurcation, pitchfork bifurcation, flip bifurcation and the Neimark-Sacker bifurcation.