STABILITY REGIONS FOR COUPLED HILLS EQUATIONS

In this paper we extend well-known results for one Hill's equation and present the stability analysis of two coupled Hill's equations for wh ich the general theory is not readily available. Approximate expressio ns are derived in the context of peturbation theory for the boundaries between bounded and unbounded periodic solutions with frequencies ome ga = n/m (n and m are positive integers) of both linear and nonlinear coupled Mathieu equations as examples. Excellent agreement is found be tween theoretical predictions and numerical computations over large ra nges of parameter values and initial conditions. These periodic soluti ons are important because they correspond to some of the lowest-order resonances of the system and when they are stable, they turn out to ha ve 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 mechani cs.