Bifurcations from Synchrony in Homogeneous Networks: Linear Theory

Golubitsky, Martin; Lauterbach, Reiner

doi:10.1137/070704873

Cited by 41 publications

(66 citation statements)

References 21 publications

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“…Synchronous solutions may moreover undergo bifurcations with quite unusual features. Such synchrony breaking bifurcations have for example been studied in [1], [3], [6], [7], [11] and [22].…”

Section: X0mentioning

confidence: 99%

Amplified Hopf Bifurcations in Feed-Forward Networks

Rink¹,

Sanders²

2013

SIAM J. Appl. Dyn. Syst.

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In [18] the authors developed a method for computing normal forms of dynamical systems with a coupled cell network structure. We now apply this theory to one-parameter families of homogeneous feed-forward chains with 2-dimensional cells. Our main result is that Hopf bifurcations in such families generically generate branches of periodic solutions with amplitudes growing like ∼ |λ| . We explain here how these bifurcations arise generically in a broader class of feed-forward chains of arbitrary length.

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Section: X0mentioning

confidence: 99%

Amplified Hopf Bifurcations in Feed-Forward Networks

Rink¹,

Sanders²

2013

SIAM J. Appl. Dyn. Syst.

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“…Bifurcation theory has been applied to the theory of coupled cell networks in various ways (e.g. [6][7][8][9]). …”

Section: (C) Motivationmentioning

confidence: 99%

“…The results in Leite & Golubitsky [7] and Golubitsky & Lauterbach [8] relate the eigenvalues of the Jacobian J to the eigenvalues of the adjacency matrix of the network. More specifically, if μ 1 , .…”

Section: (C) Motivationmentioning

confidence: 99%

“…Thus, for k = 1, generically, the bifurcation is associated with at least an eigenvalue of the adjacency matrix and all of them have the same real part. For k ≥ 2, generically, the centre subspace at a codimension-1 synchronybreaking bifurcation is isomorphic to the real part of a generalized eigenspace of the network adjacency matrix [8]. Thus, for a general dimension k of the internal dynamics, generically, the codimension-1 bifurcation is associated with at least an eigenvalue of the adjacency matrix and all of them have the same real part.…”

Section: (C) Motivationmentioning

confidence: 99%

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On bifurcations in lifts of regular uniform coupled cell networks

Moreira

2014

Proc. R. Soc. A.

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A lift of a given network is a network that admits the first network as quotient. Assuming that a bifurcation occurs for a coupled cell system consistent with the structure of a regular network (in which all cells have the same type and receive the same number of inputs and all arrows have the same type), it is well known that some lifts exhibit new bifurcating branches of solutions. In this work, we approach this problem restricting attention to uniform networks, that is, networks that have no loops and no multiple arrows. We show that, from the bifurcation point of view, rings and their lifts are special networks. We also prove that generically there are lifts that just exhibit the bifurcating branches determined by the quotient network and, moreover, we identify all generic situations where lifts exist that may exhibit bifurcating branches that do not appear in the quotient itself.

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“…Results in [9,5] relate the eigenvalues of the Jacobian J G of a coupled cell system at X 0 with the eigenvalues of A. In order for bifurcations within the quotient network Q to lead to nonsynchronous solutions in the larger network G the center subspace of J G must be larger than the center subspace of J Q .…”

Section: Introductionmentioning

confidence: 99%

Bifurcations from regular quotient networks: A first insight

Aguiar

Dias

Golubitsky

et al. 2009

Physica D: Nonlinear Phenomena

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We consider regular (identical-edge identical-node) networks whose cells can be grouped into classes by an equivalence relation. The identification of cells in the same class determines a new network -the quotient network. In terms of the dynamics this corresponds to restricting the coupled cell systems associated with a network to flow-invariant subspaces given by equality of certain cell coordinates. Assuming a bifurcation occurs for a coupled cell system restricted to the quotient network, we ask how that bifurcation lifts to the overall space. Surprisingly, for certain networks, new branches of solutions occur besides the ones that occur in the quotient network. To investigate this phenomenon we develop a systematic method that enumerates all networks with a given quotient. We also prove necessary conditions for the existence of solutions branches not predicted by the quotient. We then apply our method to two particular quotient networks; namely, two-and three-cell bidirectional rings. We show there are no additional bifurcating solution branches when the quotient network is a two-cell bidirectional ring. However, two of the 12 five-cell networks that have the three-cell bidirectional ring as a quotient network * CMUP is supported by FCT through the programmes POCTI and POSI, with Portuguese and European Community structural funds. 1 exhibit bifurcating solutions that do not occur in the quotient itself. Thus, network architecture sometimes forces the existence of bifurcating branches in addition to the ones determined by the quotient.

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Bifurcations from Synchrony in Homogeneous Networks: Linear Theory

Cited by 41 publications

References 21 publications

Amplified Hopf Bifurcations in Feed-Forward Networks

Amplified Hopf Bifurcations in Feed-Forward Networks

On bifurcations in lifts of regular uniform coupled cell networks

Bifurcations from regular quotient networks: A first insight

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