Heteroclinic orbit

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

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

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

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

where ${\displaystyle p=t_{1}t_{2}\cdots t_{k}}$  is a sequence of symbols of length k, (of course, ${\displaystyle t_{i}\in S}$ ), and ${\displaystyle q=r_{1}r_{2}\cdots r_{m}}$  is another sequence of symbols, of length m (likewise, ${\displaystyle r_{i}\in S}$ ). The notation ${\displaystyle p^{\omega }}$  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

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

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

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

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