Heteroclinic point

A point that belongs to the domain of definition of the Hamilton function of the Hamiltonian system

(*)

such that the solution of the system (*) passing through this point asymptotically approaches some periodic solution as , and asymptotically approaches another periodic solution as . 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 (*) and the two-dimensional invariant surfaces of this system. If a two-dimensional invariant surface separates the periodic solutions and , 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 (*) (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 degrees of freedom if the periodic solutions and are replaced by invariant tori and whose respective dimensions are and , . 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)
[2]	E. Zehnder, "Homoclinic points near elliptic fixed points" Comm. Pure Appl. Math. , 26 (1973) pp. 131–182
[3]	V.K. Mel'nikov, "On the stability of the centre for time-periodic perturbations" Trans. Moscow Math. Soc , 12 (1963) pp. 1–56 Trudy Moskov. Mat. Obshch. , 12 (1963) pp. 3–52
[4a]	S. Smale, "Dynamical systems and the topological conjugacy problem for diffeomorphisms" , Proc. Internat. Congress mathematicians (Stockholm, 1962) , Inst. Mittag-Leffler (1963) pp. 490–496
[4b]	S. Smale, "Diffeomorphisms with many periodic points" S.S. Cairns (ed.) , Differential and Combinatorial Topol. (Symp. in honor of M. Morse) , Princeton Univ. Press (1965) pp. 63–80
[5]	L.P. Shil'nikov, "On a Poincaré–Birkhoff problem" Math. USSR-Sb. , 3 : 3 (1967) pp. 353–371 Mat. Sb. , 74 (116) : 3 (1967) pp. 378–397
[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 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.