Sustained Resonance in very Slowly Varying Oscillatory Hamiltonian Systems

Abstract

By formulating slowly varying oscillatory systems into Hamiltonian standard form, canonical averaging techniques can be performed automatically by symbolic manipulation programs to very high orders. For, the very slow variation considered, these high orders are required to find uniformly valid solutions. When resonance is exhibited in these systems, the original system of 2N first-order differential equations is reduced to two differential equations that embody the resonance behavior. Sustained resonance, also referred to as phase locking, occurs when the leading order frequency of the reduced system oscillates about zero for long times. The general solution procedure is illustrated, and a highly accurate asymptotic solution is found explicitly for a frequently occurring class of problems, which results when only a single harmonic of the resonance is present. This solution was not possible for the same class of problems with the usual slow time. Two test cases are considered to numerically verify all results.