5 research outputs found
Resonance bifurcations from robust homoclinic cycles
We present two calculations for a class of robust homoclinic cycles with
symmetry Z_n x Z_2^n, for which the sufficient conditions for asymptotic
stability given by Krupa and Melbourne are not optimal.
Firstly, we compute optimal conditions for asymptotic stability using
transition matrix techniques which make explicit use of the geometry of the
group action.
Secondly, through an explicit computation of the global parts of the Poincare
map near the cycle we show that, generically, the resonance bifurcations from
the cycles are supercritical: a unique branch of asymptotically stable period
orbits emerges from the resonance bifurcation and exists for coefficient values
where the cycle has lost stability. This calculation is the first to explicitly
compute the criticality of a resonance bifurcation, and answers a conjecture of
Field and Swift in a particular limiting case. Moreover, we are able to obtain
an asymptotically-correct analytic expression for the period of the bifurcating
orbit, with no adjustable parameters, which has not proved possible previously.
We show that the asymptotic analysis compares very favourably with numerical
results.Comment: 24 pages, 3 figures, submitted to Nonlinearit
Dynamics of coupled cell networks: synchrony, heteroclinic cycles and inflation
Copyright © 2011 Springer. The final publication is available at www.springerlink.comWe consider the dynamics of small networks of coupled cells. We usually assume asymmetric inputs and no global or local symmetries in the network and consider equivalence of networks in this setting; that is, when two networks with different architectures give rise to the same set of possible dynamics. Focussing on transitive (strongly connected) networks that have only one type of cell (identical cell networks) we address three questions relating the network structure to dynamics. The first question is how the structure of the network may force the existence of invariant subspaces (synchrony subspaces). The second question is how these invariant subspaces can support robust heteroclinic attractors. Finally, we investigate how the dynamics of coupled cell networks with different structures and numbers of cells can be related; in particular we consider the sets of possible “inflations” of a coupled cell network that are obtained by replacing one cell by many of the same type, in such a way that the original network dynamics is still present within a synchrony subspace. We illustrate the results with a number of examples of networks of up to six cells