Almost complete and equable heteroclinic networks

Abstract

Heteroclinic connections are trajectories that link invariant sets for an autonomous dynamical flow: these connections can robustly form networks between equilibria, for systems with flow-invariant spaces. In this paper we examine the relation between the heteroclinic network as a flow-invariant set and directed graphs of possible connections between nodes. We consider realizations of a large class of transitive digraphs as robust heteroclinic networks and show that although robust realizations are typically not complete (i.e. not all unstable manifolds of nodes are part of the network), they can be almost complete (i.e. complete up to a set of zero measure within the unstable manifold) and equable (i.e. all sets of connections from a node have the same dimension). We show there are almost complete and equable realizations that can be closed by adding a number of extra nodes and connections. We discuss some examples and describe a sense in which an equable almost complete network embedding is an optimal description of stochastically perturbed motion on the network.

References in corpus (7)

• On designing heteroclinic networks from graphs
• Designing heteroclinic and excitable networks in phase space using two populations of coupled cells
• A mechanism for switching near a heteroclinic network
• Stability in simple heteroclinic networks in ${\mathbb{R}}^4$
• Heteroclinic Dynamics of Localized Frequency Synchrony: Heteroclinic Cycles for Small Populations
• Resonance bifurcations of robust heteroclinic networks
• Construction of heteroclinic networks in ${\mathbb{R}}^4$

Cited by in corpus (6)

• Stability of cycling behaviour near a heteroclinic network model of Rock-Paper-Scissors-Lizard-Spock
• Heteroclinic Dynamics of Localized Frequency Synchrony: Stability of Heteroclinic Cycles and Networks
• Chaos in Coupled Heteroclinic Cycles and its Piecewise-Constant Representation
• Stochastic facilitation in heteroclinic communication channels
• Analysis of dynamics near heteroclinic networks in ${\mathbb{R}}^4$ with a projected map
• How many points converge to a heteroclinic network in an aperiodic way?