Computation of quasi-periodic solutions of forced dissipative systems
This paper is concerned with numerical methods for dissipative systems of forced ordinary differential equations. The problem of finding a periodic solution in the case of one forcing frequency is most conveniently formulated as a fixed point problem for a Poincaré map and the stability of the periodic solution is equivalent to the stability of the fixed point. Poincaré maps for autonomous equations are well known [\textit{J. E. Marsden} and \textit{M. F. McCracken}, The Hopf bifurcation and its applications (1976; Zbl 0346.58007)]. The author extends this method to the case of two forcing frequencies. A new type of Poincaré map is introduced; it opens also the possibility of treating quasi-periodic solutions in the case with more than two frequencies. The computer implementation of this method is discussed and interesting numerical examples are presented. By comparison to the method of generalized Fourier series of \textit{L. O. Chua} and \textit{A. Ushida} [IEEE Trans. Circuits Syst. CAS-28, 953-971 (1981; Zbl 0484.93027)] the Poincaré map one allows the treatment of the stability problem and yields a clear picture of important qualitative aspects of the quasi- periodic solutions.
scientific article; zbMATH DE number 3982431