Journal of Engineering Mathematics

, Volume 53, Issue 3–4, pp 301–336 | Cite as

Working with multiscale asymptotics

Solving Weakly nonlinear oscillator equations on long-time intervals
  • Blessing Mudavanhu
  • Robert E. O’MalleyJr
  • David B. Williams


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=O1). 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


amplitude equation averaging multiple scales oscillations renormalization singular perturbations 


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Copyright information

© Springer 2005

Authors and Affiliations

  • Blessing Mudavanhu
    • 1
  • Robert E. O’MalleyJr
    • 2
  • David B. Williams
    • 3
  1. 1.Credit Risk AnalyticsMerrill Lynch & CoNew YorkUSA
  2. 2.Department of Applied MathematicsUniversity of WashingtonSeattleUSA
  3. 3.Department of MathematicsClayton State UniversityMorrowUSA

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