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Averaging using elliptic functions: Approximation of limit cycles. (English) Zbl 0699.34032
Summary: We apply the method of averaging to first order in the small parameter \(\epsilon\) to the autonomous system \(x''+\alpha x+\beta x^ 3+\epsilon g(x,x')=0\) where we do not consider \(\beta\) 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.

MSC:
34C05 Topological structure of integral curves, singular points, limit cycles of ordinary differential equations
34C29 Averaging method for ordinary differential equations
34-04 Software, source code, etc. for problems pertaining to ordinary differential equations
Software:
MACSYMA
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