A Birkhoff--Lewis Type Theorem for Some Hamiltonian Pdes

Details A Birkhoff--Lewis Type Theorem for Some Hamiltonian Pdes A Birkhoff--Lewis Type Theorem for Some Hamiltonian Pdes

2003-10-13

arxiv.org/abs/math/0310182v1

Mathematics

Dynamical Systems

20 pages

Scientific paper

In this paper we give an extension of the Birkhoff--Lewis theorem to some semilinear PDEs. Accordingly we prove existence of infinitely many periodic orbits with large period accumulating at the origin. Such periodic orbits bifurcate from resonant finite dimensional invariant tori of the fourth order normal form of the system. Besides standard nonresonance and nondegeneracy assumptions, our main result is obtained assuming a regularizing property of the nonlinearity. We apply our main theorem to a semilinear beam equation and to a nonlinear Schr\"odinger equation with smoothing nonlinearity.

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