Abstract

Several theoretical studies deal with the stability transition curves of coupled and damped Mathieu equations utilizing numerical and asymptotic methods. In this contribution, we exploit the fact that symplectic maps describe the dynamics of Hamiltonian systems. Starting with a Hamiltonian system, a particular dissipation is introduced, which allows the extension of Hamiltonian and symplectic matrices to more general γ-Hamiltonian and μ-symplectic matrices. A proof is given that the state transition matrix of any γ-Hamiltonian system is μ-symplectic. Combined with Floquet theory, the symmetry of the Floquet multipliers with respect to a μ-circle, which is different from the unit circle, is highlighted. An attempt is made for generalizing the particular dissipation to a more general form. The methodology is applied for calculation of the stability transition curves of an example system of two coupled and damped Mathieu equations.

Keywords: Hamiltonian systems, periodic systems, Mathieu equation, parametric excitation, parametric resonance, symplectic maps
