mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, a heteroclinic network is an invariant set in the

phase space In dynamical system theory, a phase space is a space in which all possible states of a system are represented, with each possible state corresponding to one unique point in the phase space. For mechanical systems, the phase space usually ...

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. Examples include the mathematical models that describe the swinging of a ...

. It can be thought of loosely as the union of more than one

heteroclinic cycle In mathematics, a heteroclinic cycle is an invariant set in the phase space of a dynamical system. It is a topological circle of equilibrium points and connecting heteroclinic orbits. If a heteroclinic cycle is asymptotically stable, approaching t ...

. Heteroclinic networks arise naturally in a number of different types of applications, including fluid dynamics and populations dynamics. The dynamics of trajectories near to heteroclinic networks is intermittent: trajectories spend a long time performing one type of behaviour (often, close to equilibrium), before switching rapidly to another type of behaviour. This type of intermittent switching behaviour has led to several different groups of researchers using them as ways to model and understand various type of neural dynamics.

References

* * * * Dynamical systems {{mathanalysis-stub