In
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, a heteroclinic cycle is an invariant set in the
phase space
The phase space of a physical system is the set of all possible physical states of the system when described by a given parameterization. Each possible state corresponds uniquely to a point in the phase space. For mechanical systems, the p ...
of a
dynamical system
In mathematics, a dynamical system is a system in which a Function (mathematics), function describes the time dependence of a Point (geometry), point in an ambient space, such as in a parametric curve. Examples include the mathematical models ...
. It is a topological circle of
equilibrium points and connecting
heteroclinic orbits. If a heteroclinic cycle is asymptotically stable, approaching trajectories spend longer and longer periods of time in a neighbourhood of successive equilibria.
In generic dynamical systems heteroclinic connections are of high co-dimension, that is, they will not persist if parameters are varied.
Robust heteroclinic cycles
A robust heteroclinic cycle is one which persists under small changes in the underlying dynamical system. Robust cycles often arise in the presence of symmetry or other constraints which force the existence of invariant hyperplanes. A prototypical example of a robust heteroclinic cycle is the Guckenheimer–Holmes cycle. This cycle has also been studied in the context of rotating convection, and as three competing species in
population dynamics
Population dynamics is the type of mathematics used to model and study the size and age composition of populations as dynamical systems. Population dynamics is a branch of mathematical biology, and uses mathematical techniques such as differenti ...
.
See also
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Heteroclinic bifurcation
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Heteroclinic network
References
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External links
* {{scholarpedia, title=Heteroclinic cycles, urlname=Heteroclinic_cycles
Dynamical systems