On asymptotic estimates of solutions in the stability case on third-order forms (Q2735920)

A system of ordinary differential equations that has a trivial (zero) solution, is considered. It is supposed the linearized system to have \(q\) pairs of purely imaginary eigenvalues, so the system can be written in the form NEWLINE\[NEWLINE\dot\xi_k= -\omega_k \eta_k+ \sum_{r=2}^\infty X_k^{(r)} (\xi_1,\eta_1,\dots, \xi_q,\eta_q),\;\dot\eta_k= \omega_k\xi_k+ \sum_{r=2}^\infty Y_k^{(r)}(\xi_1,\eta_1,\dots, \xi_q,\eta_q), \tag{1}NEWLINE\]NEWLINE with \(k=1,2,\dots,q\), and \(X_k^{(r)}\), \(Y_k^{(r)}\) are forms of the power \(r\). The Molchanov model system NEWLINE\[NEWLINE\dot\rho_k= \rho_k\sum_{j=1}^q a_{kj} \rho_j,\quad k=1,2,\dots,q,\tag{2}NEWLINE\]NEWLINE is also considered. It is assumed that the conditions of Molchanov's stability criterion for the zero solution to equations (2) are satisfied. Then, asymptotic inequalities for the norm of solutions to system (2) are derived.