Controlling General Polynomial Networks

We consider networks of massive particles connected by non-linear springs. Some particles interact with heat baths at different temperatures, which are modeled as stochastic driving forces. The structure of the network is arbitrary, but the motion of each particle is 1D. For polynomial interactions,...

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Bibliographic Details
Published inarXiv.org
Main Authors Cuneo, Noé, Eckmann, Jean-Pierre
Format Paper Journal Article
LanguageEnglish
Published Ithaca Cornell University Library, arXiv.org 14.11.2013
Subjects
Chains
Control theory
Controllability
Mathematics - Dynamical Systems
Mathematics - Mathematical Physics
Physics - Chaotic Dynamics
Physics - Mathematical Physics
Polynomials
Stability
Topology
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Summary:We consider networks of massive particles connected by non-linear springs. Some particles interact with heat baths at different temperatures, which are modeled as stochastic driving forces. The structure of the network is arbitrary, but the motion of each particle is 1D. For polynomial interactions, we give sufficient conditions for H\"ormander's "bracket condition" to hold, which implies the uniqueness of the steady state (if it exists), as well as the controllability of the associated system in control theory. These conditions are constructive; they are formulated in terms of inequivalence of the forces (modulo translations) and/or conditions on the topology of the connections. We illustrate our results with examples, including "conducting chains" of variable cross-section. This then extends the results for a simple chain obtained in Eckmann, Pillet, Rey-Bellet (1999).
ISSN:2331-8422
DOI:10.48550/arxiv.1301.1235