zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Existence of infinitely many homoclinic orbits in Hamiltonian systems. (English) Zbl 0725.58017
We consider a Hamiltonian system in ${\bbfR}\sp{2N}$, $z'=J\nabla\sb zH(t,z)$, H being 1-periodic in time, and 0 being a hyperbolic rest point. Under global assumptions on H, we prove that there are always infinitely many orbits homoclinic to 0, i.e. such that $z(\pm \infty)=0$. Those orbits are geometrically distinct, in the following sense: $$ (x,y\text{ are geometrically distinct}),\quad \Leftrightarrow \quad (\forall n\in {\bbfZ}:\ x(.)\ne y(.-n)). $$ The approach we use here is variational, and no transversality hypothesis is needed.
Reviewer: E.Séré (Paris)

37J99Finite-dimensional Hamiltonian, Lagrangian, contact, and nonholonomic systems
Full Text: DOI EuDML
[1] Ambrosetti, A., Rabinowitz, P.: Dual variational methods in critical point theory and applications. J. Funct. Anal.14, 349--381 (1973) · Zbl 0273.49063 · doi:10.1016/0022-1236(73)90051-7
[2] Clarke, F.: Periodic solutions of Hamiltonian inclusions. J. Differ. Equations40, 1--6 (1951) · Zbl 0461.34030 · doi:10.1016/0022-0396(81)90007-3
[3] Coti Zelati, V., Ekeland, I., Séré, E.: A variational approach to homoclinic orbits in Hamiltonian systems. Math. Ann.288, 133--160 (1990) · Zbl 0731.34050 · doi:10.1007/BF01444526
[4] Ekeland, I.: Convexity Methods in Hamiltonian’ Systems. Berlin Heidelberg New York: Springer 1989 · Zbl 0742.58047
[5] Hofer, H., Wysocki, K.: First order elliptic systems and the existence of homoclinic orbits in Hamiltonian systems. Preprint Rutgers University 1989 · Zbl 0702.34039
[6] Lions, P.L.: Solutions of Hartree-Fock equations for Coulomb systems. Preperint CEREMADE n. 8607. Paris 1988 · Zbl 0693.35047
[7] Lions, P.L.: The concentration-compactness principle in the calculus of variations. Rev. Mat. Iberoam.1, 145--201 (1985) · Zbl 0704.49005
[8] Moser, J.: Stable and random motions in Dynamical Systems. Princeton: Princeton University Press 1973 · Zbl 0271.70009
[9] Poincaré, H.: Les Méthodes Nouvelles de la Mécanique Céleste. Paris: Gauthier-Villars 1899
[10] Rabinowitz, P.H.: Periodic and heteroclinic orbits for a periodic Hamiltonian system. Preprint University of Wisconsin-Madison 1988 · Zbl 0701.58023