Homoclinic orbit

In mathematics, a homoclinic orbit is a trajectory of a flow of a dynamical system which joins a saddle equilibrium point to itself. More precisely, a homoclinic orbit lies in the intersection of the stable manifold and the unstable manifold of an equilibrium.
Consider the continuous dynamical system described by the ODE
Suppose there is an equilibrium at formula_2, then a solution formula_3 is a homoclinic orbit if
If the phase space has three or more dimensions, then it is important to consider the topology of the unstable manifold of the saddle point. The figures show two cases. First, when the stable manifold is topologically a cylinder, and secondly, when the unstable manifold is topologically a Möbius strip; in this case the homoclinic orbit is called "twisted".
Discrete dynamical system.
Homoclinic orbits and homoclinic points are defined in the same way for iterated functions, as the intersection of the stable set and unstable set of some fixed point or periodic point of the system. 
We also have the notion of homoclinic orbit when considering discrete dynamical systems. In such a case, if formula_5 is a diffeomorphism of a manifold formula_6, we say that formula_7 is a homoclinic point if it has the same past and future - more specifically, if there exists a fixed (or periodic) point 
formula_8 such that
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 formula_10 is a finite set of "M" symbols. The dynamics of a point "x" is then represented by a bi-infinite string of symbols 
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
where formula_13 is a sequence of symbols of length "k", (of course, formula_14), and formula_15 is another sequence of symbols, of length "m" (likewise, formula_16). The notation formula_17 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 
with the intermediate sequence formula_19 being non-empty, and, of course, not being "p", as otherwise, the orbit would simply be formula_17.