Homoclinic point

A point $(p=p^*,q=q^*)$ which belongs to the domain of definition of the Hamilton function $H=H(p,q)$ of the Hamiltonian system

\begin{equation}\dot p=-\frac{\partial H}{\partial q},\quad\dot q=\frac{\partial H}{\partial p},\quad p=(p_1,p_2),\quad q=(q_1,q_2),\label{*}\end{equation}

and is such that the solution of the system \eqref{*} passing through it asymptotically approaches some periodic solution $T_1$ as $t\to\pm\infty$. The solution passing through the homoclinic point is itself called homoclinic.

Let $S_+$ be the surface formed by the solutions of \eqref{*} which asymptotically approach the periodic solution $T_1$ as $t\to\infty$, and let $S_1$ be the surface formed by the solutions of \eqref{*} which asymptotically approach the same solution as $t\to-\infty$. The set $S_0=S_+\cap S_-$ will then consist of homoclinic solutions. If the surfaces $S_+$ and $S_-$ intersect (or make a contact of odd order) along at least one homoclinic solution, then $S_0$ will contain infinitely many different solutions. The case in which $S_0$ contains a countable number of solutions is a structurally-stable case, i.e. $S_0$ is preserved if the function $H$ changes by a small amount. The case in which $S_0$ contains an uncountable number of different solutions is not structurally stable, i.e. degenerate. It is assumed that the periodic solution $T_1$ itself and the surfaces $S_+$ and $S_-$ are preserved if the function $H$ is changed by a small amount. This will be the case, for example, if the periodic solution $T_1$ is of hyperbolic type (cf. Hyperbolic point).

Finding homoclinic solutions of a system \eqref{*} with an arbitrary Hamilton function $H$ is a difficult task. However, if it is possible to select the variables $(p,q)$ so that the equation

$$H=H_0(p)+\epsilon H_1(p,q),$$

where $\epsilon$ is a small parameter and the function $H_1$ is $2\pi$-periodic with respect to the variable $q$, is valid, then the homoclinic solutions of \eqref{*} may be found in the form of convergent series (see reference [3] in Heteroclinic point). The existence of homoclinic solutions of \eqref{*} has been proved under much weakened restrictions on the Hamilton function of \eqref{*}.

The above definition of a homoclinic point can be applied unaltered to the case of a Hamiltonian system with $n>2$ degrees of freedom if the periodic solution $T_1$ is replaced by a $k$-dimensional invariant torus $T_k$, $0<k<n$. It is known that $(n-1)$-dimensional invariant tori have homoclinic solutions if they are of hyperbolic type.

A neighbourhood of a homoclinic solution has a complicated structure. For instance, it has been proved for the case of \eqref{*} that a countable number of periodic solutions with arbitrary large periods exists in a neighbourhood of a homoclinic solution, and that any two such solutions can be connected by a heteroclinic solution. Homoclinic solutions play an important role in the general theory of smooth dynamical systems.

See also the references to Heteroclinic point.

References

[1]	H. Poincaré, "Les méthodes nouvelles de la mécanique céleste" , 1–3 , Gauthier-Villars (1892–1899)
[2]	F. Takens, "Homoclinic points in conservative systems" Invent. Math. , 18 (1972) pp. 267–292
[3]	V.K. Mel'nikov, "On the existence of doubly asymptotic trajectories" Soviet Math. Dokl. , 14 : 4 (1973) pp. 1171–1175 Dokl. Akad. Nauk SSSR , 211 : 5 (1973) pp. 1053–1056
[4]	Z. Nitecki, "Differentiable dynamics. An introduction to the orbit structure of diffeomorphisms" , M.I.T. (1971)

Comments

The notion of a homoclinic point is not restricted to Hamiltonian dynamical systems. For a survey of recent developments see [a1].

References

[a1]	F. Takens, "Homoclinic bifurcations" A.M. Gleason (ed.) , Proc. Internat. Congress Mathematicians (Berkeley, 1986) , Amer. Math. Soc. (1987) pp. 1229–1236
[a2]	J. Guckenheimer, P. Holmes, "Non-linear oscillations, dynamical systems, and bifurcations of vector fields" , Springer (1983)