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A perturbation method based on integrating factors. (English) Zbl 0926.34043

Summary: The author shows that all integrating factors for a system of n first-order, ordinary differential equations have to satisfy a system of \(\frac{1}{2}n(n+1)\) first-order, linear partial differential equations. A perturbation method based on integrating factors is presented for problems containing a small parameter. When approximations to integrating factors have been obtained an approximation to a first integral (including an error estimate) can be given.
To show how this perturbation method works the method is applied to the Van der Pol equation, a forced Duffing equation, and a perturbed Volterra-Lotka system. Not only asymptotic approximations to first integrals are given, but it is shown how, in a rather efficient way, existence and stability of time-periodic solutions can be obtained from these approximations.

MSC:

34E10 Perturbations, asymptotics of solutions to ordinary differential equations
34C25 Periodic solutions to ordinary differential equations
34C05 Topological structure of integral curves, singular points, limit cycles of ordinary differential equations
34E05 Asymptotic expansions of solutions to ordinary differential equations
34C15 Nonlinear oscillations and coupled oscillators for ordinary differential equations
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