Homoclinic point
A point which belongs to the domain of definition of the Hamilton function
of the Hamiltonian system
![]() | (*) |
and is such that the solution of the system (*) passing through it asymptotically approaches some periodic solution as
. The solution passing through the homoclinic point is itself called homoclinic.
Let be the surface formed by the solutions of (*) which asymptotically approach the periodic solution
as
, and let
be the surface formed by the solutions of (*) which asymptotically approach the same solution as
. The set
will then consist of homoclinic solutions. If the surfaces
and
intersect (or make a contact of odd order) along at least one homoclinic solution, then
will contain infinitely many different solutions. The case in which
contains a countable number of solutions is a structurally-stable case, i.e.
is preserved if the function
changes by a small amount. The case in which
contains an uncountable number of different solutions is not structurally stable, i.e. degenerate. It is assumed that the periodic solution
itself and the surfaces
and
are preserved if the function
is changed by a small amount. This will be the case, for example, if the periodic solution
is of hyperbolic type (cf. Hyperbolic point).
Finding homoclinic solutions of a system (*) with an arbitrary Hamilton function is a difficult task. However, if it is possible to select the variables
so that the equation
![]() |
where is a small parameter and the function
is
-periodic with respect to the variable
, is valid, then the homoclinic solutions of (*) may be found in the form of convergent series (see reference [3] in Heteroclinic point). The existence of homoclinic solutions of (*) has been proved under much weakened restrictions on the Hamilton function of (*).
The above definition of a homoclinic point can be applied unaltered to the case of a Hamiltonian system with degrees of freedom if the periodic solution
is replaced by a
-dimensional invariant torus
,
. It is known that
-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 (*) 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) |
Homoclinic point. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Homoclinic_point&oldid=43474