Heteroclinic orbit

The phase portrait of the pendulum equation x' + sin x = 0. The highlighted curve shows the heteroclinic orbit from (x, x′) = (–π, 0) to (x, x′) = (π, 0). This orbit corresponds with the (rigid) pendulum starting upright, making one revolution through its lowest position, and ending upright again.

In mathematics, in the phase portrait of a dynamical system, a heteroclinic orbit (sometimes called a heteroclinic connection) is a path in phase space which joins two different equilibrium points. If the equilibrium points at the start and end of the orbit are the same, the orbit is a homoclinic orbit.

Consider the continuous dynamical system described by the ordinary differential equation

$${\dot {x}}=f(x).$$

Suppose there are equilibria at $x=x_{0},x_{1}.$ Then a solution $\phi (t)$ is a heteroclinic orbit from $x_{0}$ to $x_{1}$ if both limits are satisfied:

$${\begin{array}{rcl}\phi (t)\rightarrow x_{0}&{\text{as}}&t\rightarrow -\infty ,\\[4pt]\phi (t)\rightarrow x_{1}&{\text{as}}&t\rightarrow +\infty .\end{array}}$$

This implies that the orbit is contained in the stable manifold of $x_{1}$ and the unstable manifold of $x_{0}$.

Symbolic dynamics

By using the Markov partition, the long-time behaviour of hyperbolic system can be studied using the techniques of symbolic dynamics. In this case, a heteroclinic orbit has a particularly simple and clear representation. Suppose that $S=\{1,2,\ldots ,M\}$ is a finite set of M symbols. The dynamics of a point x is then represented by a bi-infinite string of symbols

$\sigma =\{(\ldots ,s_{-1},s_{0},s_{1},\ldots ):s_{k}\in S\;\forall k\in \mathbb {Z} \}$

A periodic point of the system is simply a recurring sequence of letters. A heteroclinic orbit is then the joining of two distinct periodic orbits. It may be written as

$p^{\omega }s_{1}s_{2}\cdots s_{n}q^{\omega }$

where $p=t_{1}t_{2}\cdots t_{k}$ is a sequence of symbols of length k, (of course, $t_{i}\in S$), and $q=r_{1}r_{2}\cdots r_{m}$ is another sequence of symbols, of length m (likewise, $r_{i}\in S$). The notation $p^{\omega }$ simply denotes the repetition of p an infinite number of times. Thus, a heteroclinic orbit can be understood as the transition from one periodic orbit to another. By contrast, a homoclinic orbit can be written as

$p^{\omega }s_{1}s_{2}\cdots s_{n}p^{\omega }$

with the intermediate sequence $s_{1}s_{2}\cdots s_{n}$ being non-empty, and, of course, not being p, as otherwise, the orbit would simply be $p^{\omega }$.

See also

• Heteroclinic connection
• Heteroclinic cycle
• Heteroclinic bifurcation
• Homoclinic orbit
• Traveling wave

References

• John Guckenheimer and Philip Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, (Applied Mathematical Sciences Vol. 42), Springer