Summary
We apply the method of averaging to first order in the small parameter ε to the autonomous system
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.
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Coppola, V.T., Rand, R.H. Averaging using elliptic functions: approximation of limit cycles. Acta Mechanica 81, 125–142 (1990). https://doi.org/10.1007/BF01176982
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DOI: https://doi.org/10.1007/BF01176982