Stability of periodic orbits of coupled-map lattices

Abstract

We consider the stability properties of spatial and temporal periodic orbits of one-dimensional coupled-map lattices. The stability matrices for them are of the block-circulant form. This helps us to reduce the problem of stability of spatially periodic orbits to the smaller orbits corresponding to the building blocks of spatial periodicity, enabling us to obtain the conditions for stability in terms of those for smaller orbits. Spatially extended nonlinear dynamical systems have recently attracted considerable attention [1-3]. This is because of their wide range of applications such as turbulence, pattern formation in natural systems, solitons, etc. They also exhibit a very rich phenomenology including a wide variety of both spatial as well as temporal periodic structures, intermittency, chaos, domain walls, kink dynamics, etc. In this Rapid Communication we address the problem of stability of spatial and temporal periodic structures. We specifically consider coupled-map lattices with nearest-neighbor couplings. Detailed numerical studies show that the coupled-map lattices give rise to a variety of rich spatial and temporal structures [ll. Consider following the general model: x, +)(i) = hpfp(x, (i))+h (f)(x, (i+ I)) +h, f,(x, (t-1)), where x, (i) is the variable associated with the ith lattice point at time t taking values in a suitably bounded phase space. The maps fp, f~, f~a re maps, such as a logistic

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“…However, periodic orbit arguments have been used to justify thermodynamic results such as non-negativity of the entropy production [27] and the Onsager relations [28] without explicitly finding any periodic orbits. Periodic orbits of spatially extended systems in the form of coupled map lattices have been considered previously [29,30], leading to block cyclic matrices similar to those observed in this paper.…”

Sudden Shifts in Ecological Systems: Intermittency and Transients in the Coupled Ricker Population Model

“…This set of maps is known to exhibit the period doubling cascade, and this cascade contains within it several classes of crises. Another key concept is the fact that the characteristics of a single map or of a small number of coupled maps are transferred to much larger coupled systems (Amritkar et al, 1991). For this reason, we focus upon a system with only two patches, arguing that the crises and transient behaviors demonstrated here will be found in much larger spatial models.…”

“…An important model of this kind is the coupled map lattice, which has also often been studied for reasons independent of its ecological significance (Amritkar et al, 1991;Amritkar and Gade, 1993;Anteneodo et al, 2003;Astakhov et al, 1995;Gade and Amritkar, 1993;Kaneko, 1992aKaneko, , 1992bKaneko, , 1993Konishi and Kaneko, 1992;Lai, 1995;Manrubia and Mikhailov, 2000;Morita, 1996;Parekh et al, 1998;Silva et al, 2001;Zhu et al, 2003;De Monte et al, 2004;Lai, 1995). For ecological models of this type, a particular map governs the local dynamics at some number of discrete locations, and the populations at each location are connected to nearby locations according to some dispersal rule.…”

Dynamics of two coupled chaotic multimode Nd:YAG lasers with intracavity frequency doubling crystal

“…The effect of coupling on the dynamics of chaotic systems have been studied for different systems like, coupled chaotic oscillators [1,2], coupled chaotic pendula [3], coupled lasers [4][5][6][7] etc. Chaos in an array of coupled systems has also been studied for coupled maps [8][9][10], coupled semiconductor lasers [11], neural networks [12] and in the Josephson junction [13]. In this paper we report the results of our numerical studies on the effect of coupling of two multimode Nd:YAG lasers with intracavity KTP crystal for frequency doubling.…”