On the functional polystability of certain essentially nonlinear nonautonomous differential systems (Q2722234)

The author studies the differential system NEWLINE\[NEWLINE\dot y=\pi(t)P(t)y+\sum_{k=2}^{m}G_k(t)y^k+R_m(t,y)\tag{1}NEWLINE\]NEWLINE with \(y=\text{col}(y_1,\ldots,y_n)\), \(\pi :[t_0,\omega) \mapsto(0,\infty)\), \(\omega \leq\infty,\) \(P(t)\in C^l([t_0,\omega) \mapsto \mathbb R^{n\times n}), \) \(\|R_m(t,y)\|=O(\|y\|^{m+\alpha})\), \(\alpha >0\), and \(G_k(t)y^k\) stands for a homogeneous \(k\)th-order \(\mathbb R^n\)-valued form with respect to \(y\) whose coefficients are continuous functions on \([t_0,\omega)\). The matrix-valued function \(P(t)\) approaches a constant matrix \(P_0\) as \(t\uparrow\omega \) and has slowly varying elements. It is assumed that the characteristic equation for \(P_0\) has exactly \(n_0\) roots with zero real parts.NEWLINENEWLINENEWLINEThe notion of functional polystability introduced in the paper requires that for a given splitting of the vector \(y\) into subvectors \(y=(y_{n_1},\ldots,y_{n_{k_0}})\) there exist positive numbers \(r_s>0, s=1,\ldots,k_0,\) and a positive function \(f:[t_0,\omega) \mapsto(0,\infty)\) such that \(f=o(1)\), \(t\uparrow\omega \), and for any \(\varepsilon >0\) one can choose such \(\delta_\varepsilon \in (0,\varepsilon)\), \(T_\varepsilon \in [T_0,\infty)\) that for any \(y_0:\|y_0\|<\delta_\varepsilon \) the solution \(y(t):y(T_\varepsilon)=y_0\) satisfies the condition NEWLINE\[NEWLINE\sum_{s=1}^{k_0}\|y_{n_s}(t)\|^{2r_s}\leq\varepsilon f(t),\quad \forall t\in [T_\varepsilon,\omega).NEWLINE\]NEWLINE NEWLINENEWLINENEWLINEFirst, system (1) is converted into a form which allows to separate the so-called critical subsystem. It is assumed that for the latter one can find an asymptotic representation of a continuous solution vanishing as \(t\uparrow\omega\). Next, by using this representation and basing on the ideas of the direct Lyapunov method, sufficient conditions under which the equation has the property of functional polystability are established.