1. Introduction

Dynamical systems represented by nonlinear or linear ordinary differential equations with periodic coefficients occur in many engineer problems (see for instance [1, 2]). The simplest example of such a system is the Mathieu equation. Most investigations in literature deal with the corresponding stability transition curves [3]. Some works analyze the stability of two coupled Mathieu equations [4–6]. In general, an asymptotic or a numerical analysis method is required for analyzing this class of systems. Perturbation techniques may lead to cumbersome expression, at least for second-order perturbation [7], and a numerical analysis may require considerable computation time. In this contribution, an extension of the theory developed in [8] is exposed in which coupled Mathieu equations are analyzed in the context of a Hamiltonian system.

The literature on Hamiltonian systems is vast. We focus on the two main references [9, 10] that are relevant for the present work. The latter focuses on linear periodic Hamiltonian systems. Although every periodic mechanical system possesses at least a small amount of dissipation, the main literature on linear Hamiltonian systems does not incorporate a dissipation. The dynamics of Hamiltonian systems can be described by symplectic maps [11]. A key fact here is that a

symplectic transformation preserves the Hamiltonian structure of the underlying dynamic system. In this work we attempt to derive an appropriate formalism for linear Hamiltonian systems incorporating a very particular dissipation. For this purpose we redefine and develop the properties of the so-called γ-Hamiltonian and μ-symplectic matrices. With the last definitions, we prove that the state transition matrix of any γ-Hamiltonian system is μ-symplectic. The relevance of the symplectic matrices or symplectic maps lies on their symmetry which allows simplifying many computations and analysis [12]. The formalism is benchmarked for two coupled and damped Mathieu equations highlighting its advantages. Due to the symmetry of the symplectic matrices, the parametric resonance zones are characterized, which allows faster computations, and with higher accuracy, of the stability transition curves. This work is an extension of the contribution presented in [8, 13].
