ARTICLE IN PRESS Chaos, Solitons and Fractals xxx (2008) xxx–xxx Contents lists available at ScienceDirect Chaos, Solitons and Fractals journal homepage: www.elsevier.com/locate/chaos Clustering and diffusion in a symplectic map lattice with non-local coupling C.F. Woellner a, S.R. Lopes a, R.L. Viana a,*, I.L. Caldas b a Departamento de Física, Universidade Federal do Paraná, Caixa Postal 19044, 81531-990 Curitiba, Paraná, Brazil b Instituto de Física, Universidade de São Paulo, Caixa Postal 66318, 05315-970 São Paulo, SP, Brazil article info abstract Article history: We study a symplectic chain with a non-local form of coupling by means of a standard map Accepted 19 August 2008 lattice where the interaction strength decreases with the lattice distance as a power-law, in Available online xxxx such a way that one can pass continuously from a local (nearest-neighbor) to a global (mean-field) type of coupling. We investigate the formation of map clusters, or spatially coherent structures generated by the system dynamics. Such clusters are found to be related to stickiness of chaotic phase-space trajectories near periodic island remnants, and also to the behavior of the diffusion coefficient. An approximate two-dimensional map is derived to explain some of the features of this connection. Ó 2008 Elsevier Ltd. All rights reserved. 1. Introduction The standard map (SM), also called Chirikov–Taylor map, besides being a paradigmatic dynamical system, describes a variety of situations of physical interest [1]. It is a two-dimensional area-preserving map p7!p0 ¼ p þðK=2pÞ sinð2pxÞ; x7!x0 ¼ x þ p0; ðmod 1Þ, where K is a measure of the non-linearity of the system, and ðp; xÞ are canonical variables which may play different roles as this map is used in physical applications. In plasma physics, the SM models the interaction between a charged particle and an electrostatic plane wave train [2,3], the behavior of drift orbits in tokamaks in the presence of drift waves [4]; and it is useful as a lowest-order description of the magnetic-field line structure in plasma confinement systems [5]. In accelerator physics, SM describes the interaction of a charged particle in a cyclotron under the influence of a periodic potential applied to a small region [6]. In optics, the SM was found to characterize the chaotic evolution of the polarization in optical fibers with modulated birefringence [7]. The quantum periodically kicked rotator, the classical limit of which is described by the SM, is one of the most intensively inves- tigated models of quantum chaos [8]. Since the SM is widely used as a paradigmatic system for studying Hamiltonian dynamics, chains of coupled SM can be considered as representative examples of volume-preserving and spatially extended systems. A physically interesting exam- ple would be a chain of coupled driven rotators, for which the coupling is modulated through a sequence of delta-pulsed kicks. One possible realization consists of an array of pendula with coils on them. If a short current pulse flows through the coils, the interaction force among the pendula will act during a short time interval and thus can be approximated with a delta function [9]. Lattices of locally coupled SM, for which each map interacts with their nearest-neighbors, have been studied as mod- els of higher-dimensional Hamiltonian systems. Unlike the isolated SM, for which the dynamics is fairly well-understood due to standard mathematical results like Poincaré-Birkhoff and KAM theorems [10], systems of coupled SM exhibit a far more complex dynamics, since KAM tori no longer divide the phase-space and thus the whole chaotic layer can be con- nected, generating phenomena like Arnold diffusion [1]. Since Arnold diffusion is extremely slow [11,12], the timescale in which we have to work turns to be too large to allow the use of perturbative approaches. Kaneko and Konishi shown * Corresponding author. E-mail address: viana@fisica.ufpr.br (R.L. Viana). 0960-0779/$ - see front matter Ó 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2008.08.026 Please cite this article in press as: Woellner CF et al., Clustering and diffusion in a symplectic map lattice with non-local coupling, Chaos, Solitons & Fractals (2008), doi:10.1016/j.chaos.2008.08.026 ARTICLE IN PRESS 2 C.F. Woellner et al. / Chaos, Solitons and Fractals xxx (2008) xxx–xxx that, in locally coupled SM lattices, anomalous diffusion exists only up to a crossover time, beyond which the diffusion becomes of a Gaussian nature [13,14]. This was explained by the fast decay of the spatial correlations over the lattice, such that the number of degrees of freedom relevant to diffusion is considerably less than the lattice size. Hence in the thermodynamic limit the system exhibits size independence and intensive behavior. Diffusion in phase-space was also investigated for globally coupled SM lattices [15,16]. There has been found extensive behavior in the limit of a strongly chaotic regime in such a way that, within this global regime, the spatial correlations are negligible and the diffusion is size-dependent. Gyorgyi and coworkers have shown that the measure of the system phase-space occupied by periodic islands versus that occupied by chaotic trajectories decays rapidly with the number of coupled maps [17]. If the number of degrees of freedom (which, in a coupled SM lattice, is twice the number of maps) is too large, the characteristics of diffusion are expected to depend largely on the measure occupied by chaotic orbits, which turns to be extremely difficult to evaluate directly. Hence we must resort to indirect methods to evaluate the measure of the phase-space occupied by chaotic orbits, like the Lyapunov spectrum and the metric (Kolmogorov–Sinai) entropy. In order to put into a unified framework both local (nearest-neighbor) and global (mean-field) coupling schemes, we introduced a non-local coupling prescription where a given site interacts with all neighbors, the corresponding strength decaying with the lattice in a power-law fashion. In this paper, we analyze the dependence of the Lyapunov spectrum and entropy on the coupling parameters, namely the strength and effective range. Our results show different scaling behav- iors with coupling strength and range, which are compatible with phase-space changes observed in the numerically obtained Poincaré maps. The dynamics generated by SM coupled lattices, although being chiefly chaotic, also presents a dependence on rem- nants of periodic islands embedded in the chaotic phase-space region [18]. This dependence is manifested in the so- called stickiness effect, which makes chaotic trajectories to spend a typically long time in the vicinity of the periodic island remnants, with a non-exponentially small probability. The sticky trajectory segments are also called flights [19], which can be studied by means of finite-time Lyapunov exponents [20]. If we consider a finite-time interval a cha- otic orbit with stickiness can be viewed as a dynamical trap, which affects the transport properties characteristic of cha- otic motion [21]. In particular, the diffusion is considerably reduced in comparison with a uniformly hyperbolic chaotic region [22]. There are other detectable effects of stickiness in lattices of coupled SM, like the formation of map clusters, which rep- resent spatially coherent regions with similar temporal evolutions. Clustering in symplectic map lattices has been de- scribed in globally coupled SM by Konishi and Kaneko [15], who also revealed their finite lifetime and fractal geometric structure. For general dissipative coupled map lattices, clustering is related to the formation of one or more synchronized patterns, and is a typical feature of global couplings [23]. In this paper, we present numerical evidence that clustering is an observable manifestation of stickiness in coupled symplectic map lattices. Cluster formation through stickiness of chaotic trajectories is closely related to the coupling properties, and this effect is enhanced as the coupling becomes more of a global nature. Some aspects of the relation between clustering and stickiness are revealed through an approximate two-dimensional map. The rest of the paper is organized as follows: in Section 2, we introduce the coupled map lattice model and some of its properties. The Lyapunov spectrum and the corresponding entropy for this model are shown in Section 3. Section 4 analyzes cluster formation and the stickiness of chaotic trajectories. Section 5 considers phase-space diffusion, whereas in Section 6 we explore some properties of an approximate two-dimensional map describing the dynamics in the vicinity of island centers, in order to explain some features of stickness. The last section contains our conclusions. 2. Map lattice with non-local coupling ðiÞ ðiÞ Let us consider a one-dimensional lattice of N sites, each of them with two state variables at discrete time n : pn and xn , where i ¼ 1; 2; ...N runs over the lattice sites. The time evolution at each site is governed by the SM, modified by the non- local coupling with the other sites, and given by Rogers and Wille [24] 0 K XN 1 ÈÉÂÃÂà ðiÞ ðiÞ pffiffiffiffiffiffiffiffiffiffi ðiþjÞ ðiÞ ðiÀjÞ ðiÞ pnþ1 ¼ pn þ a sin 2pðxn À xn Þ þ sin 2pðxn À xn Þ ; ð1Þ 2p gðaÞ j¼1 j ðiÞ ðiÞ ðiÞ xnþ1 ¼ xn þ pnþ1; ð2Þ where x 2½À1=2; þ1=2; N0 ¼ðN À 1Þ=2, with N odd; and K > 0 plays the double role of being the coupling strength and the non-linearity parameter. Hence the isolated SM does not follow from the K ! 0 limit in Eq. (1). The coupling is non-local because the summation in (1) runs over all the neighbors of a given site, but the intensity of the coupling decays with the lattice distance as a power-law, the corresponding normalization factor being 0 XN 1 gðaÞ¼2 a : ð3Þ j¼1 j Please cite this article in press as: Woellner CF et al., Clustering and diffusion in a symplectic map lattice with non-local coupling, Chaos, Solitons & Fractals (2008), doi:10.1016/j.chaos.2008.08.026 ARTICLE IN PRESS C.F.