The stability of a class of quasi-autonomous periodic systems with interval resonance
type: Item
instance of: scholarly article
title: The stability of a class of quasi-autonomous periodic systems with interval resonance
zbMATH Open document ID: 0797.58042
DOI: 10.1016/0021-8928(93)90052-N
author: A. L. Kunitsyn
author: A. S. Muratov
published in: Journal of Applied Mathematics and Mechanics
publication date: 1994-04-10T00:00:00Z
review text: Consider a \(2N + 1\)-dimensional differential system (1.1) \(\dot x = X(x,t,\varepsilon)\), where \[ X(x,t,\varepsilon) = \biggl[A_ 0 + \sum^ \infty_{k = 1} \varepsilon^ k A_ k(t)\biggr]x + X_ 0(x) + \sum^ \infty_{k = 1} \varepsilon^ k X_ k(x,t). \] Here \(x = (x_ 1,\dots,x_{2N+1})\) is the phase-variable vector, \(\varepsilon\) is a small parameter, \(X_ 0(x)\) and \(X_ k(x,t)\) are vector functions analytic in \(x\) with terms of order at least 2, \(X_ k(x,t)\) being \(\omega\)-periodic in \(t\). \(A_ 0\) and \(A_ k(t)\) are a constant and \(\omega\)-periodic \((2N + 1) \times (2N + 1)\) matrices, respectively. The authors investigate the stability of the periodic motion of (1.1) in the critical case when the linear part of (1.1) has only one zero and \(N\) pairs of purely imaginary characteristic indices. It is assumed that the system is reversible (\(t\)-invariant) in the Birkhoff sense, i.e. the following identity is satisfied: \[ MX(x,t,\varepsilon) + X(Mx,- t,\varepsilon) \equiv 0,\quad M^ 2 = E, \] \(E\) being the unit matrix. It is shown that when there is no parametric resonance, the characteristic indices of the original system are identical to first order of accuracy in \(\varepsilon\) with the eigenvalues of \(A_ 0\). Considering the cases of third- and fourth-order internal resonance, the authors give the necessary and sufficient conditions for stability, and the influence of small periodic terms on the stability is established. The obtained results are used to investigate the periodic piecewise- orbital motion of a geostationary satellite with a small reactive thrust that allows it to hover over any point of the Earth's surface.
Mathematics Subject Classification ID: 37C75
Mathematics Subject Classification ID: 34C25
Mathematics Subject Classification ID: 70M20
zbMATH DE Number: 525512
zbMATH Keywords: small parameter
zbMATH Keywords: reversible system
zbMATH Keywords: \(t\)-invariant system
zbMATH Keywords: resonance
zbMATH Keywords: quasi-autonomous systems
zbMATH Keywords: stability of periodic solutions
reviewed by: J. Kalas
MaRDI profile type: Publication
cites work: Stable and Random Motions in Dynamical Systems
cites work: Q3760862
cites work: The stability of a class of reversible systems
cites work: The reversibility of mechanical systems
cites work: Q5509213
cites work: Q3961135
cites work: Q3993419
cites work: Q3231426
full work available at URL: 0021-8928(93)90052-n
OpenAlex ID: W1975520711
description: scientific article; zbMATH DE number 525512