Homoclinic orbit

Properties

The existence of one homoclinic point implies the existence of an infinite number of them. This comes from its definition: the intersection of a stable and unstable set. Both sets are invariant by definition, which means that the forward iteration of the homoclinic point is both on the stable and unstable set. By iterating N times, the map approaches the equilibrium point by the stable set, but in every iteration it is on the unstable manifold too, which shows this property.

This property suggests that complicated dynamics arise by the existence of a homoclinic point. Indeed, Smale (1967) showed that these points leads to horseshoe map like dynamics, which is associated with chaos.

Symbolic dynamics

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

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

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 ${\displaystyle p^{\omega }}$.