Spectral method for the investigation of stability of some classes of nonautonomous differential equations (Q1603035)

The author studies the following system of the differential equations \[ \dot{x} = A(t,\varepsilon)x + f(t,x), \quad x(0,\varepsilon)= x^{0}, \tag{1} \] where \[ A(t,\varepsilon)= \sum_{0}^{\infty}A_{k}(t) \varepsilon^{k},\quad f(0,t)\equiv 0,\quad \|A(t,\varepsilon)\|\leq C,\quad t\geq 0,\quad |\varepsilon|\leq \varepsilon_{0}< 1, \] and \(A_{k}(t)\) are sufficiently smooth \(T\)-periodic matrix functions. The main results of the paper are based on the next theorem: System (1), in the case where the matrix \(A_{0}\) is constant and its spectrum \(\{\lambda_{0j}\}_{1}^{n}\) satisfies the conditions \(\lambda_{0j} -\lambda_{0k}\neq i2\pi qT^{-1}\), \(j\neq k\), \(j,k = 1,2,\ldots,n\), \(q= 0,\pm 1,\ldots,\) can be transformed by means of a nondegenerate (for sufficiently small \(|\varepsilon|\ll 1)\) change of variables to the system with almost constant and diagonal matrix \[ \dot z = Q(t,\varepsilon)z +b(z,t), \quad z(0,\varepsilon)=z^{0}, \] \[ Q(t,\varepsilon)= \Lambda(\varepsilon) +\varepsilon^{N+1}G(t,\varepsilon),\quad \Lambda(\varepsilon)=\sum_{0}^{N}\Lambda_{k} \varepsilon^{k},\quad N=0,1,2,\ldots, \] where the matrices \(\Lambda_{k}, k= 0,1,\ldots, N,\) are constant and diagonal, while the matrix \(G(t,\varepsilon)\) is uniformly bounded with respect to the norm for \(t\geq 0\) and \(|\varepsilon|\ll 1\), i.e., \(\|G(t,\varepsilon\|= O(1), t\geq 0,|\varepsilon|\leq \varepsilon_{0}\ll 1.\)