Heteroclinic orbit

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.

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.

Consider the continuous dynamical system described by the ODE

${\displaystyle {\dot {x}}=f(x)}$

Suppose there are equilibria at ${\displaystyle x=x_{0}}$ and ${\displaystyle x=x_{1}}$, then a solution ${\displaystyle \phi (t)}$ is a heteroclinic orbit from ${\displaystyle x_{0}}$ to ${\displaystyle x_{1}}$ if

${\displaystyle \phi (t)\rightarrow x_{0}\quad \mathrm {as} \quad t\rightarrow -\infty }$

and

${\displaystyle \phi (t)\rightarrow x_{1}\quad \mathrm {as} \quad t\rightarrow +\infty }$

This implies that the orbit is contained in the stable manifold of ${\displaystyle x_{1}}$ and the unstable manifold of ${\displaystyle x_{0}}$.

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 ${\displaystyle S=\{1,2,\ldots ,M\}}$ is a finite set of M symbols. The dynamics of a point x is then represented by a bi-infinite string of symbols

${\displaystyle \sigma =\{(\ldots ,s_{-1},s_{0},s_{1},\ldots ):s_{k}\in S\;\forall k\in \mathbb {Z} \}}$

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

${\displaystyle p^{\omega }s_{1}s_{2}\cdots s_{n}q^{\omega }}$

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

${\displaystyle p^{\omega }s_{1}s_{2}\cdots s_{n}p^{\omega }}$

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

See also

• Heteroclinic connection
• Heteroclinic cycle
• Heteroclinic bifurcation
• Homoclinic orbit

References

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