van Horssen, W. T. A perturbation method based on integrating factors. (English) Zbl 0926.34043 SIAM J. Appl. Math. 59, No. 4, 1427-1443 (1999). 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. Cited in 1 ReviewCited in 9 Documents 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 Keywords:integrating factor; exact differential equations; integrating vector; first integrals; perturbation method; asymptotic approximation to first integral; existence and stability of time-periodic solution PDFBibTeX XMLCite \textit{W. T. van Horssen}, SIAM J. Appl. Math. 59, No. 4, 1427--1443 (1999; Zbl 0926.34043) Full Text: DOI