Computation of quasi-periodic solutions of forced dissipative systems
type: Item
instance of: scholarly article
title: Computation of quasi-periodic solutions of forced dissipative systems
zbMATH Open document ID: 0607.65047
DOI: 10.1016/0021-9991(85)90170-6
published in: Journal of Computational Physics
publication date: 1985-01-01T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 65L05
Mathematics Subject Classification ID: 65L07
Mathematics Subject Classification ID: 34A34
Mathematics Subject Classification ID: 34C25
zbMATH DE Number: 3982431
zbMATH Keywords: dissipative systems
zbMATH Keywords: Poincaré map
zbMATH Keywords: forcing frequencies
zbMATH Keywords: numerical examples
zbMATH Keywords: comparison
zbMATH Keywords: quasi-periodic solutions
author: Christian Kaas-Petersen
reviewed by: Viorel Arnăutu
MaRDI profile type: Publication
full work available at URL: 0021-9991(85)90170-6
OpenAlex ID: W4246285347
cites work: Algorithms for computing almost periodic steady-state response of nonlinear systems to multiple input frequencies
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description: scientific article; zbMATH DE number 3982431