Homoclinics for a Hamiltonian with wells at different levels
Department or Administrative Unit
We consider a Hamiltonian equation of the form (HS) ̈q(t)=−Vq(t,q(t)) for which V has two distinct non-degenerate maxima at different levels: 0 is a local maximum and ξ=0 is an absolute maximum. Under standard non-degeneracy conditions on V, our main result is that there is a solution of (HS) homoclinic to 0. Then, supposing that another geometric condition holds, we show the existence of infinitely many solutions of (HS) homoclinic to 0 that are distinguished from one another by the number of times and regions where the solutions stay away from 0. As a corollary, we show that if there is a solution of (HS) homoclinic toξ, then there are infinitely many solutions of (HS) homoclinic to 0, distinguished by the number and position of intersections with 1/2.
Bisgard, J. (2006). Homoclinics for a Hamiltonian with wells at different levels. Calculus of Variations and Partial Differential Equations, 29(1), 1–30. https://doi.org/10.1007/s00526-006-0054-9
Calculus of Variations and Partial Differential Equations
© Springer-Verlag 2006