Homoclinics for a Hamiltonian with wells at different levels

Bisgard, James
May 2007
Calculus of Variations & Partial Differential Equations;May2007, Vol. 29 Issue 1, p1
Academic Journal
We consider a Hamiltonian equation of the form (HS) $$\ddot{q}(t) = -V_q(t, q(t))$$ for which V has two distinct non-degenerate maxima at different levels: 0 is a local maximum and $$\xi \neq 0$$ 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 to $$\xi$$ , then there are infinitely many solutions of (HS) homoclinic to 0, distinguished by the number and position of intersections with 1/2.


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