dorsal/arxiv
View SchemaLarge deviations from the thermodynamic limit in globally coupled maps
| Authors | Andreas Hamm |
|---|---|
| Categories | |
| ArXiv ID | physics/9906043 |
| URL | https://arxiv.org/abs/physics/9906043 |
| DOI | 10.1016/S0167-2789(00)00056-7 |
Abstract
Systems of a large number N of globally coupled maps have become popular as a relatively simple prototype of high-dimensional dynamics, showing many interesting and typical phenomena like synchronisation, cluster formation and multistability, and having potential applications in systems like Josephson junction arrays or in biophysical models. There exists a wealth of numerical investigations of globally coupled maps. While much progress has been made in the explanation of the macroscopic behaviour of such systems in the limit of infinite N, there is still need for a sound theory about the asymptotic behaviour of finite-N systems as N approaches infinity. This article introduces a method by which it is possible to obtain asymptotic estimates for long-term deviations from the thermodynamic limit behaviour. This method is based upon the concept of quasipotentials, originally developed by Freidlin, Wentzell, and others for describing the influence of small random perturbations on the long-term behaviour of dynamical systems. The problems of explicitly computing quasipotentials in the present context and potential approximation schemes are discussed. All the concepts described in this article are illustrated with a simple example.
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"abstract": "Systems of a large number N of globally coupled maps have become popular as a\nrelatively simple prototype of high-dimensional dynamics, showing many\ninteresting and typical phenomena like synchronisation, cluster formation and\nmultistability, and having potential applications in systems like Josephson\njunction arrays or in biophysical models. There exists a wealth of numerical\ninvestigations of globally coupled maps. While much progress has been made in\nthe explanation of the macroscopic behaviour of such systems in the limit of\ninfinite N, there is still need for a sound theory about the asymptotic\nbehaviour of finite-N systems as N approaches infinity. This article introduces\na method by which it is possible to obtain asymptotic estimates for long-term\ndeviations from the thermodynamic limit behaviour. This method is based upon\nthe concept of quasipotentials, originally developed by Freidlin, Wentzell, and\nothers for describing the influence of small random perturbations on the\nlong-term behaviour of dynamical systems. The problems of explicitly computing\nquasipotentials in the present context and potential approximation schemes are\ndiscussed. All the concepts described in this article are illustrated with a\nsimple example.",
"arxiv_id": "physics/9906043",
"authors": [
"Andreas Hamm"
],
"categories": [
"physics.data-an",
"chao-dyn",
"nlin.CD"
],
"doi": "10.1016/S0167-2789(00)00056-7",
"title": "Large deviations from the thermodynamic limit in globally coupled maps",
"url": "https://arxiv.org/abs/physics/9906043"
},
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