Stability regions for coupled Hillʹs equations

In this paper we extend well-known results for one Hillʹs equation and present the stability analysis of two coupled Hillʹs equations for which the general theory is not readily available. Approximate expressions are derived in the context of peturbation theory for the boundaries between bounded and unbounded periodic solutions with frequencies ω = n/m (n and m are positive integers) of both linear and nonlinear coupled Mathieu equations as examples. Excellent agreement is found between theoretical predictions and numerical computations over large ranges of parameter values and initial conditions. These periodic solutions are important because they correspond to some of the lowest-order resonances of the system and when they are stable, they turn out to have large regions of regular motion around them in phase space. Coupled Mathieu equations appear in numerous important physical applications, in problems of accelerator dynamics, electrohydrodynamics and mechanics.