Speaker
O. Podvigina
Description
Dynamical systems, equivariant under the action of a non-trivial
symmetry group, can possess structurally stable heteroclinic cycles.
We consider stability properties of a class of structurally stable
heteroclinic cycles in $\R^n$, which we call heteroclinic
cycles of type Z. It is well-known that a heteroclinic cycle, that is not
asymptotically stable, can attract nevertheless a positive measure set from its
neighbourhood. We call such cycles fragmentarily asymptotically
stable. Necessary and sufficient conditions
for fragmentary asymptotic stability are expressed in terms of eigenvalues
and eigenvectors of transition matrices. If all transverse eigenvalues of
linearisations near steady states that are involved in the cycle are negative, then the
condition for asymptotic stability is that the transition matrices have an eigenvalue
larger than one in absolute value.
Finally, we discuss bifurcations occurring when the conditions for asymptotic stability
or for fragmentary asymptotic stability are broken
