Heteroclinic orbit

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.

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.

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