Department of Mathematics

Dynamics of heteroclinic cycles and networks

The Dynamics of Heteroclinic Cycles and Networks Research Project is an international collaboration comprising several lecturers from The University of Auckland.

A heteroclinic cycle

Heteroclinic cycles are a special type of solution to differential equations and are known to cause intermittent and bursting behaviour in some nonlinear systems. They can occur robustly in systems with symmetries, which may be inherited from a physical system which is being modeled, or in systems with other constraints, such as the permanence of extinction in population models. Heteroclinic networks are formed from the union of several heteroclinic cycles.

The dynamics near heteroclinic cycles and networks can be very complicated. An understanding of their stability properties is important, since if a cycle or network is dynamically unstable, it will not be observed in a physical system. However, a full classification of the stability properties of even simple heteroclinic cycles has not yet been achieved.

One specific behaviour of interest in this project is that of switching between different states or sub-cycles within a network. How this behaviour changes as the network loses stability is not yet fully understood. This work is part of an ongoing collaboration with Alastair Rucklidge from The University of Leeds.

Researchers at The University of Auckland

Other collaborators