Time-reversible Hamiltonian vector fields with symplectic symmetries
The main aim of this paper is to study the existence of one-parameter families of periodic orbits passing through a typical singularity for smooth time-reversible Hamiltonian systems with respect to a symplectic involution. The dynamical systems analyzed are given by vector fields presenting a \(1:-1\) resonance at the equilibrium (in two degrees of freedom). It is observed that only two distinct nontrivial symplectic involutions can arise. Moreover, these symplectic involutions force the system to have a nonpositive Hessian, and the linearized system to be a \(1:-1\) resonant. The approach used consists in using Birkhoff normal form, adapted to time-reversible systems with respect to a symplectic involution, in order to find a formal normal form which has second-order terms as a first integral. This formal normal form is used, jointly with transversality theory, restricted to the energy surfaces, to prove that the periodic solutions of the linearized system persist. The main result of the paper establishes conditions on systems of two-degrees-of-freedom for which there are two one-paremeter families of reversible periodic orbits passing through the equilibrium point. The paper is organized as follows. Reversible systems are briefly reviewed in Section 2.1. Next, some results concerning Hamiltonian systems (Section 2.2) and the anti-invariant theory (Section 2.3) are given. In Section 3.1, time-reversible Hamiltonian systems of one-degree-of-freedom are studied, classifying generic two-parameter families of reversible Hamiltonian vector fields of one-degree-of-freedom. Finally, in Section 3.2, time-reversible Hamiltonian systems with two degrees of freedom are studied, and the main result of the paper is obtained.
scientific article; zbMATH DE number 2145902