Heteroclinic orbit

File:Heteroclinic orbit in pendulum phaseportrait.png

The phase portrait of the pendulum equation xTemplate:Thinsp″ + 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 $\varphi (t)$ is a heteroclinic orbit from $x_0$ to $x_1$ if both limits are satisfied: $$\begin{array}{rcl}\varphi (t)\to x_0& \text{as}& t\to -\mathrm{\infty },\\ \varphi (t)\to x_1& \text{as}& t\to +\mathrm{\infty }.\end{array}$$

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 $S=\{1,2,\dots ,M\}$ is a finite set of M symbols. The dynamics of a point x is then represented by a bi-infinite string of symbols

$\sigma =\{(\dots ,s_{-1},s_0,s_1,\dots ):s_k\in S\mathrm{\forall }k\in \mathbb{\mathbb{Z}}\}$

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