Dynamics on Networks
Heteroclinic cycles and networks can occur in low codimension bifurcations of vector fields and have been
intensively studied by many authors. Of particular interest are the mechanisms whereby homoclinic
bifurcations can lead to complex dynamics and chaos.
On account of the Kupka-Smale theorem, robust heteroclinic cycles and networks
can only occur when vector fields possess additional structure. This structure is invariably associated with
the presence of flow invariant subspaces. Robust heteroclinic cycles are a well-known
phenomenon in models of population dynamics, ecology and game theory based on the Lotka-Volterra and replicator equations.
They also occur widely in symmetric systems.
Over the course of the project an area of special concern has been the realization of robust heteroclinic networks in
various types of network architecture. In particular in the general class of semilinear feedback systems (which includes
Lotka-Volterra systems and many equivariant systems) and in the class of identical coupled cell systems.
In Heteroclinic networks in homogeneous and heterogeneous identical cell systems
(Journal of Nonlinear Science, 25(3) (2015), 779-813) results
were proved that enabled realization of heteroclinic networks in coupled homogeneous and heterogeneous
systems of identical cells. Also considered models for network dynamics that allow variation in the
number of inputs to identical cells: Additive input structure.
In
Patterns of Desynchronization and Resychronization in Heteroclinic Networks,
Nonlinearity, 30(2) (2017), 516-557,
general results were obtained linking the theories of semilinear feedback systems and identical coupled cell systems.
This paper provides many examples of robust heteroclinic cycles and networks in identical coupled cell systems.
From the point of view of applications,
there has been recent interest in robust heteroclinic cycles that appear in neural microcircuits
where they give nonlinear models with `winnerless competition' -
there is a local competition between different
states but not necessarily a global winner. We refer to the papers listed above for more details and specific references.
email: mikefield@gmail.com
Professor Mike Field
Department of Mechanical Engineering
University of California
Santa Barbara, CA 93106
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