ON THE MULTIPLICITY OF PERIODIC ORBITS AND HOMOCLINICS NEAR CRITICAL ENERGY LEVELS OF HAMILTONIAN SYSTEMS IN R-4

Autor(es)

de Paulo, Naiara V. ^[1] ; Salomao, Pedro A. S. ^[2]

Afiliação do(s) autor(es)

^[1] Univ Fed Santa Catarina, Dept Matemat, Rua Joao Pessoa 2514, Blumenau, SC - Brazil

^[2] Univ Sao Paulo, Dept Matemat, Inst Matemat & Estat, Rua Matao 1010, BR-05508090 Sao Paulo, SP - Brazil

• Tipo de documento: Artigo Científico
• Fonte: TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY; v. 372, n. 2, p. 859-887, JUL 15 2019.
• Citações Web of Science: 0

Resumo

We study two-degree-of-freedom Hamiltonian systems. Let us assume that the zero energy level of a real-analytic Hamiltonian function H : R-4 -> R contains a saddle-center equilibrium point lying in a strictly convex sphere-like singular subset S-0 subset of H-1 (0). From previous work {[}Mem. Amer. Math. Soc. 252 (2018)] we know that for any small energy E > 0, the energy level H-1 (E) contains a closed 3-ball S-E in a neighborhood of S-0 admitting a singular foliation called 2 - 3 foliation. One of the binding orbits of this singular foliation is the Lyapunoff orbit P-2,P- E contained in the center manifold of the saddle-center. The other binding orbit lies in the interior of S-E and spans a one parameter family of disks transverse to the Hamiltonian vector field. In this article we show that the 2-3 foliation forces the existence of infinitely many periodic orbits and infinitely many homoclinics to P-2,P- E in S-E. Moreover, if the branches of the stable and unstable manifolds of P-2,P- E inside S-E do not coincide, then the Hamiltonian flow on S-E has positive topological entropy. We also present applications of these results to some classical Hamiltonian systems. (AU)

• Processo FAPESP: 16/25053-8 - Dinâmica e geometria em baixas dimensões
• Beneficiário: André Salles de Carvalho
• Modalidade de apoio: Auxílio à Pesquisa - Temático
• Processo FAPESP: 14/08113-1 - Sistemas de seções transversais na mecânica clássica
• Beneficiário: Naiara Vergian de Paulo
• Modalidade de apoio: Bolsas no Brasil - Pós-Doutorado