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Essential critical points of linking type and solutions of minimal period to superquadratic Hamiltonian systems. (English) Zbl 0776.58030

The authors generalize a result of I. Ekeland and H. Hofer [Invent. Math. 81, 155-188 (1985; Zbl 0594.58035)] on the existence of periodic solutions with prescribed minimal period for convex Hamiltonian systems. They assume the Hamiltonian \(H\in C^ 2(\mathbb{R}^{2N},\mathbb{R})\) to be strictly convex in an annulus \(K=\{z\in\mathbb{R}^{2N}:k_ 1<| z|<k_ 2\}\). They also require certain growth conditions on \(H\) and \(H'\) which depend on \(k_ 2/k_ 1\). Then they obtain the existence of periodic solutions \(z(t)\in K\) of \(J\dot z=H'(z)\) with minimal period \(T\) for each \(T\) in an interval which depends on \(k_ 1\), \(k_ 2\) and on the growth of \(H\) and \(H'\). For the proof the authors study the usual functional \(F:V=H^ 1(S^ 1,\mathbb{R}^{2N})\to\mathbb{R}\) restricted to finite-dimensional subspaces \(V_ n\) of \(V\). The solutions are obtained as limits of so-called essential critical points \(z_ n^{(m)}\) of a perturbation \(F_ n^{(m)}\) of \(F| V_ n\). The \(z_ n^{(m)}\) are nondegenerate, their Morse indices can be estimated, hence the minimal period can be controlled.

MSC:

37G99 Local and nonlocal bifurcation theory for dynamical systems
58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
34C25 Periodic solutions to ordinary differential equations
37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems

Citations:

Zbl 0594.58035
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References:

[1] Benci, V.; Fortunato, D., A Birkhoff-Lewis type result for non-autonomous differential equations, (Partial Differential Equations (Rio de Janeiro, 1986), Vol. 1324 (1988), Springer: Springer Berlin), 85-96, Lecture Notes in Mathematics
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