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Robust Heteroclinic Cycles and Networks - Claire Postlethwaite

PhD thesis, University of Cambridge, 2005.


A heteroclinic cycle is a topological circle of connecting orbits
between at least two saddle-type equilibria. In generic
(non-symmetric) dynamical systems, heteroclinic cycles are of high
codimension. However, if a dynamical system
contains flow-invariant subspaces, the connecting orbits can be contained
within these subspaces. The heteroclinic
cycle is then robust to perturbations of the system that preserve
the invariance of these subspaces.

In my dissertation, I study the dynamics and bifurcations of robust
heteroclinic cycles and networks. I consider continuous-time
dynamical systems, written as a set of ODEs that
commute with a symmetry group. The invariant subspaces arise
as a consequence of the equivariance of the dynamics; isotropy is 
conserved along trajectories.

The first part of my dissertation gives an example of complicated dynamics
near a heteroclinic network.
Heteroclinic networks are invariant sets consisting of more than one
heteroclinic cycle.
The simplest can be formed by coupling together
multiple copies of the same heteroclinic cycle in a symmetric
manner. This coupling can form new symmetric heteroclinic cycles which
are subsets of the network.
The specific network I have studied is a pair of coupled 
Guckenheimer--Holmes cycles.
I examine a particular set of
trajectories which occur when each sub-cycle within the network is 
transversally
unstable. Trajectories spend increasingly long times near successive
sub-cycles, but the $\omega$-limit set
of each trajectory is the entire network. This cycling
behaviour can occur in a regular or irregular fashion.

The next chapter gives an example of a codimension-two resonant
bifurcation from a heteroclinic cycle. I study a particular cycle which 
has a pair of complex
conjugate eigenvalues at each equilibrium. Resonant bifurcations are 
usually associated with the birth or death of a
nearby (long period) periodic orbit and as such
can occur in a subcritical or supercritical manner. I investigate
the codimension-two point that separates these two cases. The twisting 
of the orbits on the
stable manifold due to the complex eigenvalues causes a
further sequence of saddle-node bifurcations of the periodic orbits to 
occur. There is a
complex yet ordered structure of these bifurcations around the
codimension-two point.

The final chapter of my dissertation is of a more
general nature and generalises some stability theorems and
classifications which have been given in previous literature.
I consider a large class of robust
homoclinic cycles and derive necessary and sufficient stability 
conditions which improve on
previously known sufficient conditions. I also extend previous
work on transverse bifurcations from heteroclinic cycles.