The
phase portrait of the
pendulum equation x' + sin x = 0. The highlighted curve shows the heteroclinic orbit from (x, x′) = (–π, 0) to (x, x′) = (π, 0). This orbit corresponds with the (rigid) pendulum starting upright, making one revolution through its lowest position, and ending upright again.
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 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 is a sequence of symbols of length k, (of course, ), and is another sequence of symbols, of length m (likewise, ). The notation 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 being non-empty, and, of course, not being p, as otherwise, the orbit would simply be .
John Guckenheimer and
Philip Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, (Applied Mathematical Sciences Vol. 42), Springer