Nilpotent Hopf Bifurcations in coupled cell systems

Elmhirst, Toby, and Golubitsky, Martin (2006) Nilpotent Hopf Bifurcations in coupled cell systems. SIAM Journal Applied Dynamical Systems, 5 (2). pp. 205-251.

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Network architecture can lead to robust synchrony in coupled systems and, surprisingly, to codimension one bifurcations from synchronous equilibria at which the associated Jacobian is nilpotent. We prove three theorems concerning nilpotent Hopf bifurcations from synchronous equilibria to periodic solutions, where the critical eigenvalues have algebraic multiplicity two and geometric multiplicity one, and discuss these results in the context of three different networks in which the bifurcations occur generically. Phenomena stemming from these bifurcations include multiple periodic solutions, solutions that grow at a rate faster than the standard λ12, and solutions that grow slower than the standard λ12. These different bifurcations depend on the network architecture and, in particular, on the flow-invariant subspaces that are forced to exist by the architecture.

Item ID: 9163
Item Type: Article (Research - C1)
ISSN: 1536-0040
Keywords: Hopf bifurcation; coupled cells; nonsemisimple normal form
Funders: NSF
Date Deposited: 17 Mar 2010 05:49
FoR Codes: 01 MATHEMATICAL SCIENCES > 0102 Applied Mathematics > 010202 Biological Mathematics @ 50%
01 MATHEMATICAL SCIENCES > 0101 Pure Mathematics > 010109 Ordinary Differential Equations, Difference Equations and Dynamical Systems @ 50%
SEO Codes: 97 EXPANDING KNOWLEDGE > 970101 Expanding Knowledge in the Mathematical Sciences @ 100%
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