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Multiple homoclinic orbits for a class of conservative systems. (English) Zbl 0806.58018
A class of autonomous, second order classical hamiltonians \(H = {1\over 2} | p |^ 2 - {1\over 2}| q|^ 2 + W(q)\), \((p,q) \in \mathbb{R}^ n\), is considered, where essentially \(a| q|^ \alpha \leq W(q) \leq b| q|^ \alpha\) for some \(\alpha > 2\). The authors show that given a “pinching condition” \(b/a < 2^{(\alpha - 2)/2}\) there exist at least two orbits homoclinic to the origin. Moreover, if \(W\) is even, then there are indeed \(n\) such homoclinics. They also have results for intermediate multiplicities.
Since \(H\) is independent of time, this multiplicity has nothing to do with the splitting of stable and unstable manifolds (which would lead to infinitely many homoclinics). The homoclinics here could all be degenerate.
The proof is variational. It is possible to project out the radial direction so that it suffices to find critical points of the functional \(F(u) = {1\over 2} | u |^ 2_{H^{1,2}} - \int^ \infty_{- \infty} W(u)\) on the sphere \(\{| u| = 1\}\). Lusternik-Schnirelman category and a comparison argument then yield the minimum number of critical points, respectively homoclinic orbits.

MSC:
37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems
58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
70H05 Hamilton’s equations
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