Ukr. Mat. Zh. - 2018. - 70, № 1. - pp. 115-129
This paper deals with forced frequency locking, i.e., the behavior of periodic solutions to autonomous differential equations under the influence of small periodic forcings. We show that, although the forcings are allowed to be discontinuous (e.g., step-function-like) or even distributional (e.g., Dirac-function-like), the forced frequency locking happens as in the case of smooth forcings, and we derive formulas for the locking cones and for the asymptotic phases as in the case of smooth forcings.
Robustness of exponential dichotomies of boundary-value problems for general first-order hyperbolic systems
Ukr. Mat. Zh. - 2013. - 65, № 2. - pp. 236-251
We examine the robustness of exponential dichotomies of boundary-value problems for general linear first-order one-dimensional hyperbolic systems. It is assumed that the boundary conditions guarantee an increase in the smoothness of solutions in a finite time interval, which includes reflection boundary conditions. We show that the dichotomy survives in the space of continuous functions under small perturbations of all coefficients in the differential equations.
Ukr. Mat. Zh. - 2005. - 57, № 7. - pp. 922–945
Using methods of perturbation theory, we investigate the global behavior of trajectories on a toroidal attractor and in its neighborhood for a system of differential equations that arises in the study of synchronization of oscillations in the mathematical model of an optical laser.