Ignat'ev A. O.
Ukr. Mat. Zh. - 2015. - 67, № 11. - pp. 1569-1573
We consider a nonautonomous system of ordinary differential equations. It is supposed that this system has a periodic solution. We establish the lower bound for the period of this solution.
On the existence of a lyapunov function as a quadratic form for impulsive systems of linear differential equations
Ukr. Mat. Zh. - 2010. - 62, № 11. - pp. 1451–1458
A system of linear differential equations with pulse action at fixed times is considered. We obtain sufficient conditions for the existence of a positive-definite quadratic form whose derivative along the solutions of differential equations and whose variation at the points of pulse action are negative-definite quadratic forms regardless of the times of pulse action.
Ukr. Mat. Zh. - 2008. - 60, № 10. - pp. 1317–1325
A system of ordinary differential equations with impulse effects at fixed moments of time is considered. This system admits the zero solution. Sufficient conditions of the equiasymptotic stability of the zero solution are obtained.
Ukr. Mat. Zh. - 2003. - 55, № 8. - pp. 1035-1043
We prove that the sufficient conditions for the asymptotic stability of impulsive systems obtained by Gurgula and Perestyuk are also necessary conditions.
On the stability of invariant sets of a system of autonomous differential equations under constantly acting perturbations
Ukr. Mat. Zh. - 1999. - 51, № 9. - pp. 1287–1291
We consider a system of ordinary autonomous differential equations that has an invariant set. We obtain sufficient conditions for the stability of this system under constantly acting perturbations.
Ukr. Mat. Zh. - 1993. - 45, № 7. - pp. 932–941
A nonautonomous system of ordinary differential equations is considered. If this system admits a uniformly asymptotically stable integral set, then a neighborhood of this set contains a function similar to the Lyapunov function.