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Working with multiscale asymptotics

Solving Weakly nonlinear oscillator equations on long-time intervals

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Abstract

This paper surveys, compares and updates techniques to obtain the asymptotic solution of the weakly nonlinear oscillator equation ÿ + y +ε f(y, \(\dot{y}\)) =0 as ε → 0 and for corresponding first-order vector systems. The solutions found by the regular perturbation method generally feature resonance and so break down as t → ∞. The classical methods of averaging and multiple scales eliminate such secular behavior and provide asymptotic solutions valid for time intervals of length t=O(ε−1). The renormalization group method proposed by Chen et al. [Phys. Rev. E 54 (1996) 376–394] gives equivalent results. Several well-known examples are solved with these methods to demonstrate the respective techniques and the equivalency of the approximations produced. Finally, an amplitude-equation method is derived that incorporates the best features of all these techniques. This method is both straightforward to automate with a computer-algebra system and flexible enough to allow the forcing f to depend on the small parameter

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Authors and Affiliations

  1. Credit Risk Analytics, Merrill Lynch & Co, World Financial Center, 21st Floor, New York, NY, 10080, USA

    Blessing Mudavanhu

  2. Department of Applied Mathematics, University of Washington, Seattle, WA, 98195-2420, USA

    Robert E. O’Malley Jr

  3. Department of Mathematics, Clayton State University, 2000 Clayton State Blvd., Morrow, GA, 30260, USA

    David B. Williams

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  1. Blessing Mudavanhu
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  2. Robert E. O’Malley Jr
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  3. David B. Williams
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Correspondence to Blessing Mudavanhu.

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This paper is dedicated to Jerry Kevorkian with sincere appreciation for his long career of dedicated teaching at the University of Washington and for his substantial contributions to multiscale asymptotics

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Mudavanhu, B., O’Malley, R.E. & Williams, D.B. Working with multiscale asymptotics. J Eng Math 53, 301–336 (2005). https://doi.org/10.1007/s10665-005-9002-5

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