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A kind of structural frequency locking in generalized spatial automata.

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Title: A kind of structural frequency locking in generalized spatial automata.
Authors: Gaubert, Laurent1,2 gaubert@enib.fr, Redou, Pascal1,2
Source: Journal of Mathematical Analysis & Applications. Nov2017, Vol. 455 Issue 1, p105-126. 22p.
Subjects: Generalization, Machine theory, Synchronization, Stochastic convergence, Borel sets, Exterior differential systems
Abstract: The classical concept of synchronization is usually related to the locking of the basic frequencies and instantaneous phases of regular oscillations, and this question is addressed by studying specific kinds of coupled systems. This work presents a different point of view. We do not study the convergence of coupled systems to a synchronized behaviour, but try to answer the following question: in a population of coupled differential systems, when each cell (subsystem) exhibits a periodic behaviour, is the whole trajectory periodic? We define generalized spatial automata, with reference to continuous spatial automata, by means of coupling maps and associated measures on the set of cells: the main idea is the fact that a cell interprets its own environment via the states of the whole population and according to its own state. A natural partition of periods is such that cells belong to the same class if their trajectories share a common period. We demonstrate that in a general case where cells belong to an a priori unstructured set, and their trajectories evolve in possibly distinct Banach spaces, the set of classes of periods is generally countable. In particular, when the set of cells is endowed with a Borelian structure, all the cells necessarily share a common period. [ABSTRACT FROM AUTHOR]
Copyright of Journal of Mathematical Analysis & Applications is the property of Academic Press Inc. and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)
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  Data: <searchLink fieldCode="JN" term="%22Journal+of+Mathematical+Analysis+%26+Applications%22">Journal of Mathematical Analysis & Applications</searchLink>. Nov2017, Vol. 455 Issue 1, p105-126. 22p.
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  Data: <searchLink fieldCode="DE" term="%22Generalization%22">Generalization</searchLink><br /><searchLink fieldCode="DE" term="%22Machine+theory%22">Machine theory</searchLink><br /><searchLink fieldCode="DE" term="%22Synchronization%22">Synchronization</searchLink><br /><searchLink fieldCode="DE" term="%22Stochastic+convergence%22">Stochastic convergence</searchLink><br /><searchLink fieldCode="DE" term="%22Borel+sets%22">Borel sets</searchLink><br /><searchLink fieldCode="DE" term="%22Exterior+differential+systems%22">Exterior differential systems</searchLink>
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  Data: The classical concept of synchronization is usually related to the locking of the basic frequencies and instantaneous phases of regular oscillations, and this question is addressed by studying specific kinds of coupled systems. This work presents a different point of view. We do not study the convergence of coupled systems to a synchronized behaviour, but try to answer the following question: in a population of coupled differential systems, when each cell (subsystem) exhibits a periodic behaviour, is the whole trajectory periodic? We define generalized spatial automata, with reference to continuous spatial automata, by means of coupling maps and associated measures on the set of cells: the main idea is the fact that a cell interprets its own environment via the states of the whole population and according to its own state. A natural partition of periods is such that cells belong to the same class if their trajectories share a common period. We demonstrate that in a general case where cells belong to an a priori unstructured set, and their trajectories evolve in possibly distinct Banach spaces, the set of classes of periods is generally countable. In particular, when the set of cells is endowed with a Borelian structure, all the cells necessarily share a common period. [ABSTRACT FROM AUTHOR]
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  Data: <i>Copyright of Journal of Mathematical Analysis & Applications is the property of Academic Press Inc. and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract.</i> (Copyright applies to all Abstracts.)
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        Value: 10.1016/j.jmaa.2017.05.052
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        Text: English
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      – SubjectFull: Synchronization
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      – SubjectFull: Stochastic convergence
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      – SubjectFull: Borel sets
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      – SubjectFull: Exterior differential systems
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              Text: Nov2017
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