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.

Consider the continuous dynamical system described by the ODE
::$\dot x=f(x)$
Suppose there are equilibria at $x=x_0$ and $x=x_1$, then a solution $\phi(t)$ is a heteroclinic orbit from $x_0$ to $x_1$ if
::$\phi(t)\rightarrow x_0\quad \mathrm\quad t\rightarrow-\infty$
and
::$\phi(t)\rightarrow x_1\quad \mathrm\quad t\rightarrow+\infty$

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=\$ 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

Dynamical systems