Resonance in a Weakly Nonlinear System with Slowly Varying Parameters

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Abstract
We survey multiple‐variable expansion procedures appropriate for nonlinear systems in resonance using the model of two coupled weakly nonlinear oscillators with either constant or slowly varying frequencies. In the autonomous problem we show that an n‐variable expansion (where n depends on the order of accuracy desired) yields uniformly valid results. We also consider the problem of passage through resonance for the nonautonomous problem and describe the solution by constructing a sequence of three expansions. The solution before resonance is developed as a generalized multiple‐variable expansion and is matched with an inner expansion valid during resonance. This latter is then matched with a postresonance solution and determines it completely. Numerical integrations are used to substantiate the theoretical results. The dominant effect of passage through resonance is shown to be the excitation of a higher‐order oscillation beyond resonance. Contrary to the claim in a recent work, the total action of the system does not remain constant if one accounts for the leading perturbation terms in the postresonance solution. Instead, the total action goes from one constant value to another.