Stability of Coupled and Damped Mathieu Equations Utilizing Symplectic Properties

Several theoretical studies deal with the stability transition curves of the Mathieu equation. A few others present numerical and asymptotic methods to describe the stability of coupled Mathieu equations. However, sometimes the averaging and perturbation techniques deal with cumbersome computations, and the numerical methods spend considerable resources and computation time. This contribution extends the definition of linear Hamiltonian systems to periodic Hamiltonian systems with a particular dissipation. This leads naturally to a generalization of symplectic matrices, to mu-symplectic matrices. This definition enables an efficient way for calculating the stability transition curves of coupled Mathieu equations.