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Interior symmetry and local bifurcation in coupled cell networks
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UNSPECIFIED (2004) Interior symmetry and local bifurcation in coupled cell networks. DYNAMICAL SYSTEMS-AN INTERNATIONAL JOURNAL, 19 (4). pp. 389-407. doi:10.1080/14689360512331318006 ISSN 1468-9367.
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Official URL: http://dx.doi.org/10.1080/14689360512331318006
Abstract
A coupled cell system is a network of dynamical systems, or 'cells', coupled together. Such systems can be represented schematically by a directed graph whose nodes correspond to cells and whose edges represent couplings. A symmetry of a coupled cell system is a permutation of the cells and edges that preserves all internal dynamics and all couplings. It is well known that symmetry can lead to patterns of synchronized cells, rotating waves, multirhythms, and synchronized chaos. Recently, the introduction of a less stringent form of symmetry, the 'symmetry groupoid', has shown that global group-theoretic symmetry is not the only mechanism that can create such states in a coupled cell system. The symmetry groupoid consists of structure-preserving bijections between certain subsets of the cell network, the input sets. Here, we introduce a concept intermediate between the groupoid symmetries and the global group symmetries of a network: 'interior symmetry'. This concept is closely related to the groupoid structure, but imposes stronger constraints of a group-theoretic nature. We develop the local bifurcation theory of coupled cell systems possessing interior symmetries, by analogy with symmetric bifurcation theory. The main results are analogues for 'synchrony-breaking' bifurcations of the Equivariant Branching Lemma for steady-state bifurcation, and the Equivariant Hopf Theorem for bifurcation to time-periodic states.
Item Type: | Journal Article | ||||
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Subjects: | Q Science > QA Mathematics T Technology > TJ Mechanical engineering and machinery |
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Journal or Publication Title: | DYNAMICAL SYSTEMS-AN INTERNATIONAL JOURNAL | ||||
Publisher: | TAYLOR & FRANCIS LTD | ||||
ISSN: | 1468-9367 | ||||
Official Date: | December 2004 | ||||
Dates: |
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Volume: | 19 | ||||
Number: | 4 | ||||
Number of Pages: | 19 | ||||
Page Range: | pp. 389-407 | ||||
DOI: | 10.1080/14689360512331318006 | ||||
Publication Status: | Published |
Data sourced from Thomson Reuters' Web of Knowledge
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