Heteroclinic orbit explained

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 $$\backslash dot\; x\; =\; f(x).$$Suppose there are equilibria at

x = x _{0,x} _{1.} Then a solution \phi(t) is a heteroclinic orbit from x _{0} to x _{1} if both limits are satisfied:

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 ,\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 \sigma = \{(\ldots,s _{-1} ,s _{0,s} _{1,\ldots)} : s _{k} \in S \forall k \in 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} … s _{n} q ^{\omega}

where

p = t _{1} t _{2} … t _{k} is a sequence of symbols of length k, (of course, t _{i\in} S ), and q = r _{1} r _{2} … 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} … s _{n} p ^{\omega}

with the intermediate sequence

s _{1} s _{2} … 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