Estimates of solutions in critical cases of stability theory
Here, the system of autonomous differential equations NEWLINE\[NEWLINE\dot x= Ax+X(x,y),\;\dot y=By+Y(x,y),\;x\in \mathbb{R}^n,\;y\in\mathbb{R}^m,\tag{1}NEWLINE\]NEWLINE is considered where \(A\) is a \(k\otimes k\)-matrix with pure imaginary eigenvalues, \(B\) is an \(m\otimes m\)-matrix whose eigenvalues have negative real parts, \(X\) and \(Y\) are analytic vector-functions of \(x,y\). Using the known Pliss theorem [Izv. Akad. Nauk SSSR, Ser. Mat. 28, No. 6, 1297-1324 (1964; Zbl 0131.31505)] the author studies the problem of finding power-law estimates for the convergence of solutions to (1) to zero in some Lyapunov critical cases.NEWLINENEWLINENEWLINEMoreover, the author studies the infinitesimal order of admissible perturbations for the ``critical'' subsystem \(\dot x=Ax+ X(x,f(x))\) where \(f(x)\) defines an invariant manifold to (1). In the particular case of two single zero eigenvalues of \(A\) the author obtains sharp constants and exponents in the estimates for the convergence of solutions to zero, gives the orders of admissible stability-preserving perturbations, and describes the phase portrait of the considered system.
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