For codimension one equivariant networks, the non-wandering dynamics is conjugate to a topological Markov chain. An example in codimension two of a resonant homoclinic loop to an equilibrium with ID3 symmetry is presented.

Abstract:

We consider heteroclinic networks with a transitive group action. Suppose the equilibria in the network have real leading eigenvalues. For generic codimension one equivariant networks of this type we show that the non-wandering dynamics is conjugate to a topological Markov chain. The situation is more complicated in codimension two and we demonstrate a novel example to illustrate the difficulties and interesting phenomena that occur. This example is of a resonant homoclinic loop to an equilibrium with D_3 symmetry.

Keywords:

Homoclinic; Non-wandering dynamics; Symbolic Dynamics; Symmetry; Markov chain.

Lecture #13229

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