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Multiple homoclinics in a Hamiltonian system with asymptotically or super linear terms. (English) Zbl 1104.70013
Summary: This paper is concerned with homoclinic orbits in the Hamiltonian system z ˙=𝒥H z (t,z), where H is periodic in t with H z (t,z)=L(t)z+R z (t,z), R z (t,z)=o(|z|) as z0. We find a condition on the matrix-valued function L to describe the spectrum of operator -(𝒥d/dt+L), so that a proper variational formulation is presented. Supposing R z is asymptotically linear as |z| and symmetric in z, we obtain infinitely many homoclinic orbits. We also treat the case where R z is super linear as |z| with assumptions different from those studied previously, and prove existence and multiplicity of homoclinic orbits. Our arguments are based on some recent information on strongly indefinite functionals in critical point theory.
70K44Homoclinic and heteroclinic trajectories (nonlinear dynamics)
70H05Hamilton’s equations