On asymptotic estimates of solutions in the stability case on third-order forms.(Russian)Zbl 0982.34043

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 $\dot\xi_k= -\omega_k \eta_k+ \sum_{r=2}^\infty X_k^{(r)} (\xi_1,\eta_1,\dots, \xi_q,\eta_q),\;\dot\eta_k= \omega_k\xi_k+ \sum_{r=2}^\infty Y_k^{(r)}(\xi_1,\eta_1,\dots, \xi_q,\eta_q), \tag{1}$ with $$k=1,2,\dots,q$$, and $$X_k^{(r)}$$, $$Y_k^{(r)}$$ are forms of the power $$r$$. The Molchanov model system $\dot\rho_k= \rho_k\sum_{j=1}^q a_{kj} \rho_j,\quad k=1,2,\dots,q,\tag{2}$ 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.

MSC:

 34D20 Stability of solutions to ordinary differential equations

Keywords:

Lyapunov function; stable solution