Minimal P-symmetric period problem of first-order autonomous Hamiltonian systems

Abstract

Let P ∈ Sp(2n) satisfying P^k = I_2n. We consider the minimal Psymmetric period problem of the autonomous nonlinear Hamiltonian system \dot{x}(t)=J{H}^{\prime }(x(t)). For some symplectic matrices P, we show that for any τ>0, the above Hamiltonian system possesses a kτ periodic solution x with kτ being its minimal P-symmetric period provided H satisfies Rabinowitz’s conditions on the minimal period conjecture, together with that H is convex and H(Px) = H(x).

Key words： Maslov P-index; relative Morse index; minimal P-symmetric period; Hamiltonian system

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