Heteroclinic Cycles and Wreath Product Symmetries

Dias, Ana Paula S.; Dionne, B.; Stewart, Ian

doi:10.1142/9789812794543_0009

Cited by 5 publications

(11 citation statements)

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“…Another bifurcation will be a resonance that causes the network to lose stability and results in bifurcation of periodic or other attractors from the network. Unlike the heteroclinic networks considered in [15] we do not expect the networks we consider to appear in primary bifurcations from a trivial state -the symmetries we require for robustness do not act transitively on the set of cells.…”

Section: Discussionmentioning

confidence: 93%

“…The system is memoryless if all possible transitions are memoryless. We expect such a system will be truly memoryless only in the singular limit of low noise ζ → 0; nonetheless, later sections suggest that a system can be very close to memoryless in that corrections to (15) may be asymptotically small in ζ. Note also that a particular transition will necessarily be memoryless if π p,q = 1, though this does not necessarily imply that all other transitions are memoryless.…”

Section: Statistical Properties Of Trajectories Near a Realised Networkmentioning

confidence: 94%

“…For example, heteroclinic networks with the structure of "odd graphs" [8] can be found in systems of N = 2k + 1 oscillators, and heteroclinic ratchets can be found in less symmetric coupled systems [20]. Up to now there seems to be no way to easily design a coupled dynamical system that realises a given directed network as a robust heteroclinic attractor; the closest approaches we are aware of are [16,15,30,1]. The last of these specifically considers the design of heteroclinic networks using small numbers of coupled cells by a process of "inflation" of cells in a smaller network but it is not clear how to use this to design an arbitrary heteroclinic network.…”

Section: Introductionmentioning

confidence: 99%

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On designing heteroclinic networks from graphs

Ashwin

Postlethwaite

2013

Physica D: Nonlinear Phenomena

114

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Robust heteroclinic networks are invariant sets that can appear as attractors in symmetrically coupled or otherwise constrained dynamical systems. These networks may have a very complicated structure that is poorly understood and determined to a large extent by the constraints and dimension of the system. As these networks are of great interest as dynamical models of biological and cognitive processes, it is useful to understand how particular graphs can be realised as robust heteroclinic networks that are attracting. This paper presents two methods of realizing arbitrarily complex directed graphs as robust heteroclinic networks for flows generated by ODEs---we say the ODEs {\em realise} the graphs as heteroclinic networks between equilibria that represent the vertices. Suppose we have a directed graph on $n_v$ vertices with $n_e$ edges. The "simplex realisation" embeds the graph as an invariant set of a flow on an $(n_v-1)$-simplex. This method realises the graph as long as it is one- and two-cycle free. The "cylinder realisation" embeds a graph as an invariant set of a flow on a $(n_e+1)$-dimensional space. This method realises the graph as long as it is one-cycle free. In both cases we find the graph as an invariant set within an attractor, and discuss some illustrative examples, including the influence of noise and parameters on the dynamics. In particular we show that the resulting heteroclinic network may or may not display "memory" of the vertices visited.Comment: 31 pages, 16 figure

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Section: Discussionmentioning

confidence: 93%

Section: Statistical Properties Of Trajectories Near a Realised Networkmentioning

confidence: 94%

Section: Introductionmentioning

confidence: 99%

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On designing heteroclinic networks from graphs

Ashwin

Postlethwaite

2013

Physica D: Nonlinear Phenomena

114

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“…Finally, we briefly discuss the construction of homoclinic and heteroclinic cycles in R n with symmetry groups Γ = Z n Z n 2 : details can be found in [132], and the reference [108] contains further information on their appearance in local bifurcations. A very useful tool is the invariant-sphere theorem which we explain first.…”

Section: Lemma 52 ([187]mentioning

confidence: 99%

Homoclinic and Heteroclinic Bifurcations in Vector Fields

Homburg

Sandstede

2010

Handbook of Dynamical Systems

160

229

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An overview of homoclinic and heteroclinic bifurcation theory for autonomous vector fields is given. Specifically, homoclinic and heteroclinic bifurcations of codimension one and two in generic, equivariant, reversible, and conservative systems are reviewed, and results pertaining to the existence of multi-round homoclinic and periodic orbits and of complicated dynamics such as suspended horseshoes and attractors are stated. Bifurcations of homoclinic orbits from equilibria in local bifurcations are also considered. The main analytic and geometric techniques such as Lin's method, Shil'nikov variables and homoclinic center manifolds for analyzing these bifurcations are discussed. Finally, a few related topics, such as topological moduli, numerical algorithms, variational methods, and extensions to singularly perturbed and infinitedimensional systems, are reviewed briefly.

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“…There is a rich theory for heteroclinic cycles and networks when H is a transitive subgroup of S n (and G acts irreducibly on R). We refer to Field (2007) for details and more references as well as to Dias et al (2000) who investigate more general linear actions by wreath products.…”

Section: Heteroclinic Cycles and Networkmentioning

confidence: 99%