Perturbation Techniques for Oscillatory Systems with Slowly Varying Coefficients

Abstract

This is an expository paper on perturbation techniques and results for a general system of first order equations that model various weakly nonlinear oscillatory motions with slowly varying parameters. Sample problems from three application areas are derived and then transformed to a standard form. Asymptotic solutions and adiabatic invariants are calculated using the method of near identity averaging transformations. Equivalent results are also derived using the method of multiple scales. When certain divisors in the solution tend to zero, one can isolate the resonant behavior in a reduced problem of lower order. The solution of this reduced problem is discussed in detail for the case of transient resonance, and qualitatively for the case of sustained resonance.