Sustained Resonance for a Nonlinear System with Slowly Varying Coefficients

J. Kevorkian [SIAM J. Appl. Math., 20 (1971), pp. 364–373; 26 (1974), pp. 638–669] studied resonance for a spinning reentry vehicle using a model system of ordinary differential equations with slowly varying coefficients. He and L. Lewin [SIAM J. Appl. Math., 35 (1978), pp. 738–754] gave formal multiple-time-scale expansions and numerical results to give a description of a mechanism for capture in sustained resonance. J. Sanders [SIAM J. Math. Anal., 10 (1979), pp. 1220–1243] studied these equations more rigorously using the method of averaging, but still could not prove the existence of sustained resonance. This paper continues the study of these equations using higher order averaging and the Melnikov method and shows rigorously that capture in sustained resonance does take place for some initial conditions. The Melnikov method measures the opening of a saddle connection for a small perturbation in terms of an integral. Since it has usually been applied to perturbations which depend periodically on time, the derivation of the integral for systems with slowly varying coefficients is included.