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Heteroclinic networks, chaos and switching

By: Isabel S. Labouriau
From: Centro de Matemática da Universidade do Porto
At: Complexo Interdisciplinar, Anfiteatro
[2006-02-22] 11:30

A heteroclinic network for an ordinary differential equation consists of a finite number of flow invariant subsets (the nodes of the network) connected by trajectories. In the simplest example we have equilibria of the equation and trajectories that approach these equiibria when time tends to ±∞. Usually these network are not persistent: a small modification of the equations may destroy the connections. When the equations have some symmetry, however, if the connections take place inside the ``mirrors' of the symmetry, the networks may persist, as long as we still have symmetry.

In this talk I will discuss how to use symmetry to obtain equations with these networks. I will discuss one specific example, where the flow near the network may be shown to shadow all the possible paths on the network and also to contain the suspension of infinitely many horseshoes. In this way we find, near the network, a flow-invariant set that is a complex suspension of horseshoes; a kind of Cantor spaghetti in the shape of the network. When symmetry is broken the network may be destroyed but the flow-invarant set remains.