The stability of a class of quasi-autonomous periodic systems with interval resonance
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.
scientific article; zbMATH DE number 525512