Heteroclinic point

A point $(p=p^*,q=q^*)$ that 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}

such that the solution of the system \eqref{*} passing through this point asymptotically approaches some periodic solution $T_1$ as $t\to\infty$, and asymptotically approaches another periodic solution $T_1'$ as $t\to-\infty$. The solution itself, which passes through the heteroclinic point, is called a heteroclinic solution.

There is a connection between the heteroclinic solutions of the system \eqref{*} and the two-dimensional invariant surfaces of this system. If a two-dimensional invariant surface separates the periodic solutions $T_1$ and $T_1'$, there is no heteroclinic solution joining these periodic solutions. In many cases the converse is true. In the non-degenerate case, in a neighbourhood of a homoclinic solution (cf. Homoclinic point) there exists an infinite sequence of periodic solutions any two of which may be joined by a heteroclinic solution. A neighbourhood of a contour consisting of a finite number of periodic and heteroclinic solutions of the system \eqref{*} (a so-called homoclinic cycle) has a structure that in many respects resembles that of a homoclinic solution.

The above definition of a heteroclinic point may be applied practically unchanged to the case of a Hamiltonian system with $n>2$ degrees of freedom if the periodic solutions $T_1$ and $T_1'$ are replaced by invariant tori $T_k$ and $T'_{k'}$ whose respective dimensions are $k$ and $k'$, $0<k,k'<n$. Heteroclinic solutions play an important part in the study of instability in Hamiltonian systems with number of degrees of freedom higher than two and in the theory of structurally-stable dynamical systems (cf. Rough system).

References

[1]	H. Poincaré, "Les méthodes nouvelles de la mécanique céleste" , 1–3 , Gauthier-Villars (1892–1899)
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[6a]	V.M. Alekseev, "Quasirandom dynamical systems I" Math. USSR-Sb. , 5 : 1 (1968) pp. 73–128 Mat. Sb. , 76 (118) : 1 (1968) pp. 72–134
[6b]	V.M. Alekseev, "Quasirandom dynamical systems II" Math. USSR-Sb. , 6 : 4 (1968) pp. 505–560 Mat. Sb. , 77 (119) : 4 (1968) pp. 545–601
[6c]	V.M. Alekseev, "Quasirandom dynamical systems III" Math. USSR-Sb. , 7 : 1 (1969) pp. 1–43 Mat. Sb. , 78 (120) : 1 (1969) pp. 3–50

Comments

The above notion of a heteroclinic (homoclinic) point is meaningful for arbitrary continuous-time dynamical systems (not necessarily Hamiltonian). It can also be defined for dynamical systems with discrete time: Let $f$ be a diffeomorphism on a manifold. Then a heteroclinic (homoclinic) point is any point which is in the intersection of the stable manifold of one invariant point and the unstable manifold of another (the same) invariant point.