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Heteroclinic orbit

Path between equilibrium points in a phase space

1=(''x'', ''x''′) = (π, 0)}}. This orbit corresponds with the (rigid) pendulum starting upright, making one revolution through its lowest position, and ending upright again.

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 \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: \begin{array}{rcl} \phi(t) \rightarrow x_0 &\text{as}& t \rightarrow -\infty, \[4pt] \phi(t) \rightarrow x_1 &\text{as}& t \rightarrow +\infty. \end{array}

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 \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

:p^\omega s_1 s_2 \cdots s_n q^\omega

where p= t_1 t_2 \cdots t_k is a sequence of symbols of length k, (of course, t_i\in S), and q = r_1 r_2 \cdots 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 \cdots s_n p^\omega

with the intermediate sequence s_1 s_2 \cdots s_n being non-empty, and, of course, not being p, as otherwise, the orbit would simply be p^\omega.

References

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

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dynamical-systems

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