Low-Dimensional Chaos in Populations of Strongly-Coupled Noisy Maps

Monte, Silvia De; d’Ovidio, Francesco; Mosekilde, Erik; Chaté, Hugues

doi:10.1143/ptps.161.27

Cited by 4 publications

(4 citation statements)

References 32 publications

(23 reference statements)

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“…Following these preimages and dealing with the singularity due to the superstable point f ′ (y) = 0 make the system practically inaccessible by numerical means. This singularity is tamed by adding noise to the system as in Equation ( 3), or by studying a heterogeneous system with site-dependent local parameters [20,21], which do not destroy collective dynamics but provides a useful situation to elucidate the nature of the collective behavior [9][10][11].…”

Section: Correspondence To Perron-frobenius Lyapunov Modesmentioning

confidence: 99%

“…We start with a rather simple regime of collective chaos under strong coupling, where dynamical variables tend to synchronize to the chaotic dynamics of the uncoupled logistic map, but are weakly scattered by microscopic chaos and noise [10,11,30]. Specifically, we set the local logistic parameter at a = 1.57, which corresponds to the one-band chaos regime, and in the present section K = 0.28 and σ = 0.1, unless otherwise specified.…”

Section: Correspondence To Perron-frobenius Lyapunov Modesmentioning

confidence: 99%

“…Such non-trivial collective behavior (NTCB) is now known to be quite generic, being observed whether interactions are local or global, time variable is discrete or continuous [5][6][7][8], and evolution is deterministic or noisy [9][10][11]. One of the most interesting features of NTCB is that microscopic chaos can coexist with macroscopic evolutions of different instability -periodic, quasi-periodic, and even chaotic in the case of global coupling [4,12].…”

Section: Introductionmentioning

confidence: 99%

“…Emergence of collective behavior in large dynamical systems is a striking example of situations where microscopic interactions lead to non-trivial time-dependent macroscopic behavior, in contrast with equilibrium systems: a collection of dynamical units is in general not self-averaging, and hence macroscopic observables may show a variety of temporal evolutions even in the presence of microscopic chaos and in the absence of synchronization [1][2][3][4]. Such non-trivial collective behavior (NTCB) is now known to be quite generic, being observed whether interactions are local or global, time variable is discrete or continuous [5][6][7][8], and evolution is deterministic or noisy [9][10][11]. One of the most interesting features of NTCB is that microscopic chaos can coexist with macroscopic evolutions of different instability -periodic, quasi-periodic, and even chaotic in the case of global coupling [4,12].…”

Section: Introductionmentioning

confidence: 99%

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Collective Lyapunov modes

Takeuchi¹,

Chaté²

2013

J. Phys. A: Math. Theor.

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Abstract. We show, using covariant Lyapunov vectors in addition to standard Lyapunov analysis, that there exists a set of collective Lyapunov modes in large chaotic systems exhibiting collective dynamics. Associated with delocalized Lyapunov vectors, they act collectively on the trajectory and hence characterize the instability of its collective dynamics. We further develop, for globally-coupled systems, a connection between these collective modes and the Lyapunov modes in the corresponding PerronFrobenius equation. We thereby address the fundamental question of the effective dimension of collective dynamics and discuss the extensivity of chaos in presence of collective dynamics.

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Section: Correspondence To Perron-frobenius Lyapunov Modesmentioning

confidence: 99%

Section: Correspondence To Perron-frobenius Lyapunov Modesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Collective Lyapunov modes

Takeuchi¹,

Chaté²

2013

J. Phys. A: Math. Theor.

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Globally coupled chaotic maps and demographic stochasticity

Kessler

Shnerb

2010

Phys. Rev. E

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The affect of demographic stochasticity of a system of globally coupled chaotic maps is considered. A two-step model is studied, where the intra-patch chaotic dynamics is followed by a migration step that coupled all patches; the equilibrium number of agents on each site, N , controls the strength of the discreteness-induced fluctuations. For small N (large fluctuations) a period-doubling cascade appears as the coupling (migration) increases. As N grows an extremely slow dynamic emerges, leading to a flow along a one-dimensional family of almost period 2 solutions. This manifold become a true solutions in the deterministic limit. The degeneracy between different attractors that characterizes the clustering phase of the deterministic system is thus the N → ∞ limit of the slow dynamics manifold.PACS numbers: 87.23. Cc , 64.70.qj, 05.45.Xt, The dynamics of coupled chaotic maps have attracted a lot of interest in the last decades, following the pioneering works of Kaneko [1,2]. A substantial part of the study is focused around the paradigmatic model of globally coupled maps, where many fundamental results like mutual synchronization, dynamical clustering and glassy behavior were demonstrated [3]. The universal character of the chaotic dynamics makes the coupled maps model relevant to many phenomena, ranging from neural systems and human body rhythms to coupled lasers and cryptography [4].Here we consider the effect of demographic stochasticity (shot noise) on various phases of a globally coupled system. This problem emerges naturally while applying the theory to spatially extended ecologies.Many old [5] and recent [6] experiments suggest that the well-mixed dynamics of simple ecosystems (single species or victim-exploiter system) are extinction-prone, and that the system acquires stability only due to its spatial structure, a result supported also by numerical simulations of many models [7,8]. The extended (spatial) system survives due to the possibility of migration among spatial patches. This migration should be large enough to allow for recolonization of empty patches be emigrants. On the other hand [9], too much migration is also dangerous, as it leads to global synchronization, in which case the system acts essentially as a single, wellmixed patch, with its vulnerability to extinction.The globally coupled system, which obeys,is a natural and popular framework to discuss the dynamics of so-called meta-populations [10] on various patches with migration between the patches [9]. Here, s i is the population density on the i-th site and F is the chaotic map. ν is the migration parameter (the chance of an individual agent to leave its site) and 1 ≤ i ≤ L, where L is the number of patches. To address the problem of extinction properly, however, it is necessary to account for the discrete nature of the population and the absorbing character of the zero population state. This is achieved by the addition of demographic (shot) noise to the system. Clearly, if the local populations are all large, this effect is tiny. Howeve...

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Lyapunov Analysis Captures the Collective Dynamics of Large Chaotic Systems

2009

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We show, using generic globally coupled systems, that the collective dynamics of large chaotic systems is encoded in their Lyapunov spectra: most modes are typically localized on a few degrees of freedom, but some are delocalized, acting collectively on the trajectory. For globally coupled maps, we show, moreover, a quantitative correspondence between the collective modes and some of the so-called Perron-Frobenius dynamics. Our results imply that the conventional definition of extensivity must be changed as soon as collective dynamics sets in.

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Low-Dimensional Chaos in Populations of Strongly-Coupled Noisy Maps

Cited by 4 publications

References 32 publications

Collective Lyapunov modes

Collective Lyapunov modes

Globally coupled chaotic maps and demographic stochasticity

Lyapunov Analysis Captures the Collective Dynamics of Large Chaotic Systems

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