Heteroclinic orbit

Path between equilibrium points in a phase space
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.

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

x ˙ = f ( x ) . {\displaystyle {\dot {x}}=f(x).}
Suppose there are equilibria at x = x 0 , x 1 . {\displaystyle x=x_{0},x_{1}.} Then a solution ϕ ( t ) {\displaystyle \phi (t)} is a heteroclinic orbit from x 0 {\displaystyle x_{0}} to x 1 {\displaystyle x_{1}} if both limits are satisfied:
ϕ ( t ) → x 0 as t → − ∞ , ϕ ( t ) → x 1 as t → + ∞ . {\displaystyle {\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 {\displaystyle x_{1}} and the unstable manifold of x 0 {\displaystyle 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 , … , M } {\displaystyle 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

σ = { ( … , s − 1 , s 0 , s 1 , … ) : s k ∈ S ∀ k ∈ Z } {\displaystyle \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 ω s 1 s 2 ⋯ s n q ω {\displaystyle p^{\omega }s_{1}s_{2}\cdots s_{n}q^{\omega }}

where p = t 1 t 2 ⋯ t k {\displaystyle p=t_{1}t_{2}\cdots t_{k}} is a sequence of symbols of length k, (of course, t i ∈ S {\displaystyle t_{i}\in S} ), and q = r 1 r 2 ⋯ r m {\displaystyle q=r_{1}r_{2}\cdots r_{m}} is another sequence of symbols, of length m (likewise, r i ∈ S {\displaystyle r_{i}\in S} ). The notation p ω {\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

p ω s 1 s 2 ⋯ s n p ω {\displaystyle p^{\omega }s_{1}s_{2}\cdots s_{n}p^{\omega }}

with the intermediate sequence s 1 s 2 ⋯ s n {\displaystyle s_{1}s_{2}\cdots s_{n}} being non-empty, and, of course, not being p, as otherwise, the orbit would simply be p ω {\displaystyle p^{\omega }} .

See also

References

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