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Heteroclinic point

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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.

How to Cite This Entry:
Heteroclinic point. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Heteroclinic_point&oldid=43473
This article was adapted from an original article by V.K. Mel'nikov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article