# Kolesov A. Yu.

### New methods for the investigation of periodic solutions in ring systems of unidirectionally coupled oscillators

Ukr. Mat. Zh. - 2013. - 65, № 1. - pp. 82-102

We consider special systems of ordinary differential equations, namely, ring systems of unidirectionally coupled oscillators. A new method is developed for the investigation of the problem of the existence and stability of periodic solutions for this class of systems. The specific feature of this approach is the use of certain auxiliary delay systems for the determination of cycles and for the analysis of their properties. The proposed method is illustrated by a specific example.

### On one bifurcation in relaxation systems

Kolesov A. Yu., Mishchenko E. F., Rozov N. Kh.

Ukr. Mat. Zh. - 2008. - 60, № 1. - pp. 63–72

We establish conditions under which, in three-dimensional relaxation systems of the form $$\dot{x} = f(x, y, \mu),\quad, \varepsilon\dot{y} = g(x, y),\quad x= (x_1, x_2) \in {\mathbb R}^2,\quad y\in{\mathbb R },$$ where $0 < ε << 1, |μ| << 1, ƒ, g ∈ C_{∞}$ the so-called “blue-sky catastrophe” is observed, i.e., there appears a stable relaxation cycle whose period and length tend to infinity as μ tends to a certain critical value μ*(ε), μ*(0) 0 = 0.

### Parametric bufferness in systems of parabolic and hyperbolic equations with small diffusion

Kolesov A. Yu., Mishchenko E. F., Rozov N. Kh.

Ukr. Mat. Zh. - 1998. - 50, № 1. - pp. 22–35

We investigate the problem of parametric excitation of oscillations in systems of parabolic and hyperbolic equations with small coefficient of diffusion. We establish the phenomenon of parametric bufferness, i.e., the existence of an arbitrary fixed number of stable periodic solutions for a proper choice of the parameters of equations.