Acta Mechanica

, Volume 81, Issue 3–4, pp 125–142 | Cite as

Averaging using elliptic functions: approximation of limit cycles

  • V. T. Coppola
  • R. H. Rand
Contributed Papers


We apply the method of averaging to first order in the small parameter ε to the autonomous system
$$x'' + \alpha x + \beta x^3 + \varepsilon g\left( {x, x'} \right) = 0$$
where we do not consider β as small. This involves perturbing off of Jacobian elliptic functions, rather than off of trigonometric functions as is usually done. The resulting equations involve integrals of elliptic functions which are evaluated using a program written in the computer algebra system MACSYMA. The results are applied to the problem of approximating limit cycles in the above differential equation.


Differential Equation Dynamical System Fluid Dynamics Small Parameter Transport Phenomenon 
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Copyright information

© Springer-Verlag 1990

Authors and Affiliations

  • V. T. Coppola
    • 1
  • R. H. Rand
    • 1
  1. 1.Department of Theoretical and Applied MechanicsCornell UniversityIthacaNYUSA

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