Synchronous chaos in coupled map lattices with small-world interactions

Gade, Prashant M.; Hu, Chin–Kun

doi:10.1103/physreve.62.6409

Cited by 231 publications

(110 citation statements)

References 16 publications

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“…Among those, automata epidemics simulations [13] and Web-browsing studies [14] have revealed the importance of shortcuts. Numerical work on synchronization of Kuramoto oscillators [3], discrete maps [15] and Hodgkin-Huxley neurons [16] has shown improved SW synchronizability, as intuitively expected. However, these numerical examples are not generic, and fail to provide insight into how the SW property influences the dynamics.…”

Section: Pacs Numbersmentioning

confidence: 99%

Synchronization in Small-World Systems

2002

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We quantify the dynamical implications of the small-world phenomenon. We consider the generic synchronization of oscillator networks of arbitrary topology, and link the linear stability of the synchronous state to an algebraic condition of the Laplacian of the graph. We show numerically that the addition of random shortcuts produces improved network synchronizability. Further, we use a perturbation analysis to place the synchronization threshold in relation to the boundaries of the small-world region. Our results also show that small-worlds synchronize as efficiently as random graphs and hypercubes, and more so than standard constructive graphs. PACS numbers:Recently, Watts and Strogatz [1] showed that the addition of a few long-range shortcuts to an otherwise locally connected lattice (the "pristine world") produces a sharp reduction of the average distance between arbitrary nodes. The ensuing semi-random lattice was denoted a small-world (SW) because the sudden appearance of short paths occurs early on, while the system is still relatively localized. This concept has wide appeal: the SW property seems to be a quantifiable characteristic of many real-world structures [1, 2, 3, 4], both human generated (social networks, WWW, power grid), or of biological origin (neural and biochemical networks).A spur of ongoing research [2] has concentrated on static and combinatoric properties [5,6,7,8,9, 10] of a tractable SW model [1,11]. Monasson [11] considered the SW effect on the distribution of eigenvalues of the connectivity matrix (the graph Laplacian) which specifies the coupling between nodes-a relevant topic for polymer networks [12]. However, despite their central role in real-world networks, there are fewer studies of dynamical processes taking place on SW lattices. Among those, automata epidemics simulations [13] In this paper, we explicitly link the SW addition of random shortcuts to the synchronization of networks of coupled dynamical systems. This is an example of dynamics on networks-leaving aside the distinct problem of evolution of networks here. By using a generic synchronization formulation [17,18] to factor out the connectivity of the network, we identify the synchronization threshold with an algebraic condition of the graph Laplacian. Through numerics and analysis, we quantify how the SW scheme improves the synchronizability of the pristine world, mainly as a result of the steep increase of the first-non-zero eigenvalue (FNZE). The synchronization threshold is found to lie in the SW region [3, 13], but does not coincide with its onset-it can in fact be linked to the effective randomization that ends SW. Within this framework, we show that the synchronization efficiency of semi-random SW networks is higher than standard deterministic graphs, and comparable to both fully random and ideal constructive graphs.Consider n identical dynamical systems (placed at the nodes of a graph) that are linearly and symmetrically coupled (as represented by the edges of the undirected graph) with global coupling strengt...

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Section: Pacs Numbersmentioning

confidence: 99%

Synchronization in Small-World Systems

2002

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“…Bishop & Atmanspacher [2004]). For this reason, several authors have directed their attention to the analysis of the stability in CMLs (e.g., Mackey & Milton [1995], Belykh et al [2000], Gade & Hu [2000], Gelover-Santiago et al As a common feature of the (so far) few studies of asynchronous updating, it has been reported that it facilitates the synchronization and stabilization of CMLs decisively. In particular, Mehta & Sinha [2000] demonstrated that the dynamics at individual lattice cells is strongly synchronized by coupling among cells.…”

Section: Stabilizationmentioning

confidence: 99%

Stabilization of causally and non-causally coupled map lattices

Atmanspacher

Scheingraber

2005

Physica A: Statistical Mechanics and its Applications

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Two-dimensional coupled map lattices have global stability properties that depend on the coupling between individual maps and their neighborhood. The action of the neighborhood on individual maps can be implemented in terms of "causal" coupling (to spatially distant past states) or "non-causal" coupling (to spatially distant simultaneous states). In this contribution we show that globally stable behavior of coupled map lattices is facilitated by causal coupling, thus indicating a surprising relationship between stability and causality. The influence of causal versus non-causal coupling for synchronous and asynchronous updating as a function of coupling strength and for different neighborhoods is analyzed in detail.

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“…Lots of existing work on synchronization are conducted on chaos (e.g. Gade & Hu (2000)), coupled maps (e.g. Jalan & Amritkar (2003)), scale-free or small-world networks (e.g.…”

Section: Decentralized Synchronization With Adaptive Controlmentioning

confidence: 99%