Dynamics near a heteroclinic network

Aguiar, Manuela A. D.; Castro, Sofia B. S. D.; Labouriau, Isabel S.

doi:10.1088/0951-7715/18/1/019

Cited by 51 publications

(86 citation statements)

References 27 publications

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“…Networks where all the nodes are equilibria, have been studied by Postlethwaite and Dawes [21] who found trajectories that follow three cycles in a network sequentially, both regularly and irregularly; by Kirk and Silber [15] near a network with two cycles who found nearby trajectories that switch in one direction. Persistent random switching is found by Guckenheimer and Worfolk [10] and Aguiar et al [4]; noise induced switching in Armbruster et al [5].…”

Section: Introductionmentioning

confidence: 92%

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Switching near a network of rotating nodes

2009

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Abstract. We study the dynamics of a Z 2 ⊕ Z 2 -equivariant vector field in the neighbourhood of a heteroclinic network with a periodic trajectory and symmetric equilibria. We assume that around each equilibrium the linearization of the vector field has non-real eigenvalues. Trajectories starting near each node of the network turn around in space either following the periodic trajectory or due to the complex eigenvalues near the equilibria. Thus, a network with rotating nodes. The rotations combine with transverse intersections of two-dimensional invariant manifolds to create switching near the network: close to the network there are trajectories that visit neighbourhoods of the saddles following all the heteroclinic connections of the network in any given order. Our results are motivated by an example where switching was observed numerically, by forced symmetry breaking of an asymptotically stable network with O(2) symmetry.

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Section: Introductionmentioning

confidence: 92%

“…Estimates for the transition maps may be refined as in Aguiar et al [4] to show that near this network there is a suspended horseshoe with transition map described as a full shift over a countable set of symbols. The suspended horseshoe has the same shape as the network.…”

Section: Theoremmentioning

confidence: 99%

Switching near a network of rotating nodes

2009

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“…The sometimes intricate dynamics near invariant polycycles and heteroclinic networks is discussed in various papers: see, or instance, [7,133] for heteroclinic networks in R 4 that contain equilibria with complex conjugate eigenvalues and produce suspended horseshoes.…”

Section: Lemma 52 ([187]mentioning

confidence: 99%

Homoclinic and Heteroclinic Bifurcations in Vector Fields

Homburg

Sandstede

2010

Handbook of Dynamical Systems

159

228

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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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“…However this orbit space is not a manifold (it would be if the action of G were free) and its geometric, stratified structure is too difficult to compute to make this method useful in our case. We can however proceed as [1] by identifying a subgroup G 0 of G with a free action on S 3 and large enough to allow for a substantial reduction of the number of equilibria on the 3-dimenisonal manifold S 3 /G 0 . This is the aim of the next lemma.…”

Section: Quotient Networkmentioning

confidence: 99%

Bifurcation Diagrams and Heteroclinic Networks of Octagonal H-Planforms

Faye

Chossat

2012

J Nonlinear Sci

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This paper completes the classification of bifurcation diagrams for H-planforms in the Poincaré disc D whose fundamental domain is a regular octagon. An H-planform is a steady solution of a PDE or integro-differential equation in D, which is invariant under the action of a lattice subgroup Γ of U (1, 1), the group of isometries of D. In our case Γ generates a tiling of D with regular octagons. This problem was introduced as an example of spontaneous pattern formation in a model of image feature detection by the visual cortex where the features are assumed to be represented in the space of structure tensors. Under "generic" assumptions the bifurcation problem reduces to an ODE which is invariant by an irreducible representation of the group of automorphisms G of the compact Riemann surface D/Γ. The irreducible representations of G have dimension one, two, three and four. The bifurcation diagrams for the representations of dimension less than four have already been described and correspond to already well known goup actions. In the present work we compute the bifurcation diagrams for the remaining three irreducible representations of dimension four, thus completing the classification. In one of these cases, there is generic bifurcation of a heteroclinic network connecting equilibria with two different orbit types.

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Dynamics near a heteroclinic network

Cited by 51 publications

References 27 publications

Switching near a network of rotating nodes

Switching near a network of rotating nodes

Homoclinic and Heteroclinic Bifurcations in Vector Fields

Bifurcation Diagrams and Heteroclinic Networks of Octagonal H-Planforms

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