A study of self-organizing processes of nonlinear stochastic variables

Kometani, Kaoru; Shimizu, Hiroshi

doi:10.1007/bf01013146

Cited by 67 publications

(41 citation statements)

References 12 publications

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“…Here, we focus our attention on a type of collective behavior arising in globally coupled identical systems. The first evidence of collective behavior dates back to the 70's, when it was proven that a nonzero mean field may spontaneously arise in an ensemble of particles stochastically moving in a bistable potential [1]. Later, it was numerically shown that macroscopic periodic dynamics appears in coupled stochastic [2] as well as chaotic (Rössler) oscillators [3], while the first experimental evidence was found in Josephson-junction arrays [4].…”

Section: Introductionmentioning

confidence: 99%

A new approach to partial synchronization in globally coupled rotators

Mohanty¹,

Politi²

2006

J. Phys. A: Math. Gen.

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We develop a formalism to analyze the behaviour of pulse-coupled identical phase oscillators with a specific attention devoted to the onset of partial synchronization. The method, which allows describing the dynamics both at the microscopic and macroscopic level, is introduced in a general context, but then the application to the dynamics of leaky integrate-and-fire (LIF) neurons is analysed. As a result, we derive a set of delayed equations describing exactly the LIF behaviour in the thermodynamic limit. We also investigate the weak coupling regime by means of a perturbative analysis, which reveals that the evolution rule reduces to a set of ordinary differential equations. Robustness and generality of the partial synchronization regime is finally tested both by adding noise and considering different force fields.

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Section: Introductionmentioning

confidence: 99%

A new approach to partial synchronization in globally coupled rotators

Mohanty¹,

Politi²

2006

J. Phys. A: Math. Gen.

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“…This type of mean-field SDE models has been used to understand muscle contraction (see Section 5 in [13]). Other similar models have been widely studied in chemical kinetics, statistical mechanics and economics to capture cooperative behavior of a generic particle, oscillator or an agent.…”

Section: Statement Of the Problemmentioning

confidence: 99%

A Stochastic Maximum Principle for Risk-Sensitive Mean-Field Type Control

Djehiche

Tembiné

Tempone

2015

IEEE Trans. Automat. Contr.

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In this paper we study mean-field type control problems with risk-sensitive performance functionals. We establish a stochastic maximum principle (SMP) for optimal control of stochastic differential equations (SDEs) of mean-field type, in which the drift and the diffusion coefficients as well as the performance functional depend not only on the state and the control but also on the mean of the distribution of the state. Our result extends the risk-sensitive SMP (without mean-field coupling) of Lim and Zhou (2005), derived for feedback (or Markov) type optimal controls, to optimal control problems for non-Markovian dynamics which may be time-inconsistent in the sense that the Bellman optimality principle does not hold. In our approach to the risk-sensitive SMP, the smoothness assumption on the valuefunction imposed in Lim and Zhou (2005) needs not be satisfied. For a general action space a Peng's type SMP is derived, specifying the necessary conditions for optimality. Two examples are carried out to illustrate the proposed risk-sensitive mean-field type SMP under linear stochastic dynamics with exponential quadratic cost function. Explicit solutions are given for both mean-field free and mean-field models.

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“…There exists a widespread opinion that, for example, there are (at finite temperatures) no phase transitions occurring in one-dimensional (1D) systems possessing short range interactions. For the case of a global interaction, such systems can undergo a well-defined phase transition [17]. However, there are examples of 1D models with short range interactions, and very important -in presence of on-site potentials -that indeed do exhibit a true thermodynamic phase transition [11,16].…”

Section: Interaction Matricesmentioning

confidence: 99%

Can Self-Sustaining Currents Be Induced In A System Of Mesoscopic Rings?

Łuczka¹

2005

AIP Conference Proceedings

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Mesoscopic systems exhibit both quantum and classical features. A purely quantum and topological phenomenon is the occurrence of a persistent current. We analyze a collection of coaxial, mesoscopic rings which are coupled via mutual inductances. At temperatures T > 0, thermal fluctuations are taken into account. The system is described in terms of a set of Langevin equations. The unsolved problem is to find steady states in the limit of infinitely many rings (i.e. in the thermodynamic limit). The first problem is to evaluate the effective coupling constants which are determined by elements of the inverse matrix of mutual inductances. The second problem is to apply the mean-field type approximation: can it be justified? If yes, then the resulting steady states are determined by a nonlinear Fokker-Planck equation from which it follows that self-sustaining currents can be induced by interactions among rings. If no, then it is an open and much more difficult problem to identify the corresponding effective state equation.

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A study of self-organizing processes of nonlinear stochastic variables

Cited by 67 publications

References 12 publications

A new approach to partial synchronization in globally coupled rotators

A new approach to partial synchronization in globally coupled rotators

A Stochastic Maximum Principle for Risk-Sensitive Mean-Field Type Control

Can Self-Sustaining Currents Be Induced In A System Of Mesoscopic Rings?

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