Understanding the dynamics of multi--dimensional conservative dynamical systems (Hamiltonian flows or symplectic maps) is a fundamental issue of non-linear science. The Generalized ALignment Index (GALI), which was recently introduced and applied successfully for the distinction between regular and chaotic motion in Hamiltonian systems \cite{sk:6}, is an ideal tool for this purpose. In the present paper we make a first step towards the dynamical study of multi--dimensional maps, by obtaining some interesting results for a 4--dimensional (4D) symplectic map consisting of N=2 coupled standard maps \cite{Kan:1}. In particular, using the new GALI$_3$ and GALI$_4$ indices, we compute the percentages of regular and chaotic motion of the map equally reliably but much faster than previously used indices, like GALI$_2$ (known in the literature as SALI).
We analyze invariant measures of two coupled piecewise linear and everywhere expanding maps on the synchronization manifold. We observe that though the individual maps have simple and smooth functions as their stationary densities, they become multifractal as soon as two of them are coupled nonlinearly even with a small coupling. For some maps, the multifractal spectrum seems to be robust with the coupling or map parameters and for some other maps, there is a substantial variation. The origin of the multifractal spectrum here is intriguing as it does not seem to conform to the existing theory …
Unidirectionally coupled area-preserving maps with a mixed phase space may show identical synchronization in the sticky neighborhood of the regular islands. We use this fact to devise numerical procedures to control (delay and expedite) the process of synchronization in two standard maps coupled under the Pecora-Carroll coupling scheme. The delay method is based on controlled kicking of trajectories away from synchronization traps for as long as necessary. The method to expedite the process is achieved by a parameter perturbation technique which rapidly drives the chaotic trajectories to synch…
We prove a sufficient condition for synchronization for coupled one-dimensional maps and estimate the size of the window of parameters where synchronization takes place. It is shown that coupled systems on graphs with positive eigenvalues (EVs) of the normalized graph Laplacian concentrated around 1 are more amenable for synchronization. In the light of this condition, we review spectral properties of Cayley, quasirandom, power-law graphs, and expanders and relate them to synchronization of the corresponding networks. The analysis of synchronization on these graphs is illustrated with numerica…
We investigate dynamically and statistically diffusive motion in a chain of linearly coupled 2-dimensional symplectic McMillan maps and find evidence of subdiffusion in weakly and strongly chaotic regimes when all maps of the chain possess a saddle point at the origin and the central map is initially excited. In the case of weak coupling, there is either absence of diffusion or subdiffusion with $q>1$-Gaussian probability distributions, characterizing weak chaos. However, for large enough coupling and already moderate number of maps, the system exhibits strongly chaotic ($q\approx 1$) subdiffu…
We study the completely synchronized states (CSSs) of a system of coupled logistic maps as a function of three parameters: interaction strength ($\varepsilon$), range of the interaction ($α$), that can vary from first-neighbors to global coupling, and a parameter ($β$) that allows to scan continuously from non-delayed to one-time delayed dynamics. % We identify in the plane $α$-$\varepsilon$ periodic orbits, limit cycles and chaotic trajectories, and describe how these structures change with the delay. These features can be explained by studying the bifurcation diagrams of a two-dimensional no…
We study the dynamics of two conservatively coupled Hénon maps at different levels of dissipation. It is shown that the decrease of dissipation leads to changes in the parameter plane structure and scenarios of transition to chaos comparing with the case of infinitely strong dissipation. Particularly, the Feigenbaum line becomes divided into several fragments. Some of these fragments have critical points of different types, namely of C and H type, as their terminal points. Also the mechanisms of formation of these Feigenbaum line ruptures are described.
We investigate the long-term diffusion transport and chaos properties of single and coupled standard maps. We consider model parameters that are known to induce anomalous diffusion in the maps' phase spaces, as opposed to normal diffusion which is associated with Gaussian distribution properties of the kinematic variables. This type of transport originates in the presence of the so-called accelerator modes, i.e.~non-chaotic initial conditions which exhibit ballistic transport, which also affect the dynamics in their vicinity. We first systematically study the dynamics of single standard maps, …
The logistic map is a paradigmatic dynamical system originally conceived to model the discrete-time demographic growth of a population, which shockingly, shows that discrete chaos can emerge from trivial low-dimensional non-linear dynamics. In this work, we design and characterize a simple, low-cost, easy-to-handle, electronic implementation of the logistic map. In particular, our implementation allows for straightforward circuit-modifications to behave as different one-dimensional discrete-time systems. Also, we design a coupling block in order to address the behavior of two coupled maps, alt…
Effects of microscopic disorder on the collective dynamics of globally coupled maps
研究了独立加性噪声对大种群全局耦合映射同步动力学的影响。我们的分析是对Teramae和Kuramoto [Teramae和Kuramoto，Phys. Rev. E 63：036210，2001]所采取的方法的补充，Teramae和Kuramoto指出了相干性损失之前的异常标度特性。我们专注于宏观动力学，保持确定性在任何噪声水平，并从微观的不同。利用适当定义的序参量，提出了一种用近似低维系统描述集体动力学的解析方法。宏观方程的系统推导提供了人口的微观特征（单元素动力学和噪声分布）和紧急行为的属性之间的联系。由噪声引起的宏观分岔相比，那些起源于人口内的参数失配。
The problem of synchronization of coupled Hamiltonian systems exhibits interesting features due to the non-uniform or mixed nature (regular and chaotic) of the phase space. We study these features by investigating the synchronization of unidirectionally coupled area-preserving maps coupled by the Pecora-Carroll method. We find that coupled standard maps show complete synchronization for values of the nonlinearity parameter at which regular structures are still present in phase space. The distribution of synchronization times has a power law tail indicating long synchronization times for at lea…
Globally coupled maps (GCMs) are prototypical examples of high-dimensional dynamical systems. Interestingly, GCMs formed by an ensemble of weakly coupled identical chaotic units generically exhibit a hyperchaotic 'turbulent' state. A decade ago, Takeuchi et al. [Phys. Rev. Lett. 107, 124101 (2011)] theorized that in turbulent GCMs the largest Lyapunov exponent (LE), $λ(N)$, depends logarithmically on the system size $N$: $λ_\infty-λ(N)\simeq c/\ln N$. We revisit the problem and analyze, by means of analytical and numerical techniques, turbulent GCMs with positive multipliers to show that there…
Generalized synchronization in discrete maps. New point of view on weak and strong synchronization
在这封信中，我们表明，离散映射中的广义同步机制的概念需要以与流系统相同的方式进行细化[PRE，84（2011）037201]。我们已经表明，\alkor{, in the general case, when the relationship between state vectors of the interacting chaotic maps are considered,}的史前必须考虑。本文将相管方法推广到离散时间单向耦合和双向耦合系统，并利用相管方法分析了广义同步的本质。所得结果表明，将广义同步划分为弱同步和强同步也必须重新考虑。以单向耦合的Logistic映射和相互耦合的Hénon映射作为样本系统。
This paper discusses possible approaches to the escape rate in infinite lattices of weakly coupled maps with uniformly expanding repeller. It is proved that computed-via-volume rates of spatially periodic approximations grow linearly with the period size, suggesting normalized escape rate as the appropriate notion for the infinite system. The proof relies on symbolic dynamics and is based on the control of cumulative effects of perturbations within cylinder sets. A piecewise affine diffusive example is presented that exhibits monotonic decay of the escape rate with coupling intensity.
Several coupled maps models are sketched and reviewed in this short communication. First, a discrete logistic type model that was proposed for the symbiotic interaction of two species. Second, a model of many of these symbiotic species evolving together under the coupling action of a global mean field. Third, a similar model with many of these species interacting in a symbiotic way with the nearest neighboring species in different kind of networks. The phenomena of bistability and synchronization is discussed in these models. Finally, some new results concerning the last model embedded in diff…
