DYNAMICS OF ASYMMETRIC COUPLINGS OF TWO HÉNON MAPS

Abstract

In this paper, we present and analyze two asymmetric couplings of two Hénon maps. In the first case, we investigate numerically phenomena associated with the appearence, in the parameter-space, of periodic structures like Arnold tongues, which are windows of periodicity immersed in a quasiperiodic region resulting from a Naimark–Sacker bifurcation. We show that these structures organize themselves in period-adding bifurcation cascades, and that successive windows have monotonically decreasing width. In the second case, we show that when the individual Hénon maps in coupling are both chaotic, there are specific parameter values that may force the coupled maps into periodic orbits. Therefore, parameter-space regions of the coupled system where the chaotic dynamics is driven towards regular dynamics are found. In the two cases, Lyapunov exponents, bifurcation diagrams, parameter-space and phase-space plots are used to characterize the dynamics observed as parameters are changed.