---
tag: N9GYM69ZWM
kind: concept
name: Heteroclinic orbit
msc: [34C37, 37C29, 70K44]
specializes: [07TZ2Q4C1M]
sources:
  - url: https://en.wikipedia.org/wiki/Heteroclinic_orbit
same_as:
  wikidata: Q1992087
  wikipedia: Heteroclinic orbit
---

## Summary

A heteroclinic orbit is a [[trajectory|07TZ2Q4C1M]] of a dynamical system that joins two different [[equilibrium points|0N8NGKDZDZ]].
It tends to the first one in the past and to the second one in the future.
It is also called a heteroclinic connection.
Heteroclinic orbits bound regions of different behaviour in the [[phase portrait|0T3GK0XCBC]].
Their breaking under perturbation is a source of complicated dynamics.
See the [[definition|F05PSBZP8Z]].

## Explanations

### L1 human

Picture a landscape with two mountain passes and a ridge between them.
A ball that leaves the first pass and rolls exactly along the ridge to the second pass follows a heteroclinic orbit.
For the equation $x' = 1 - x^2$ on the line, the solution $x(t) = \tanh t$ is near $-1$ in the far past and near $1$ in the far future.
Its orbit, the open interval $(-1, 1)$, is a heteroclinic orbit from the equilibrium $-1$ to the equilibrium $1$.
When the starting and the final equilibrium are the same, the orbit is called homoclinic instead.