2019
Том 71
№ 1

# Stepanenko N. V.

Articles: 4
Brief Communications (Ukrainian)

### Sets of linear expansions of dynamical systems on a torus for a fixed Lyapunov function

Ukr. Mat. Zh. - 2007. - 59, № 12. - pp. 1707–1713

We consider sets of linear expansions of dynamical systems on a torus with general alternating Lyapunov function.

Article (Ukrainian)

### Alternating Lyapunov functions in the theory of linear extensions of dynamical systems on a torus

Ukr. Mat. Zh. - 2007. - 59, № 4. - pp. 488–500

We consider a series of problems connected with the application of quadratic-form Lyapunov functions to the investigation of the properties of regularity of linear extensions of dynamic systems on a torus.

Brief Communications (Ukrainian)

### On Regularity of Certain Linear Expansions of Dynamical Systems on a Torus

Ukr. Mat. Zh. - 2002. - 54, № 4. - pp. 568-574

We investigate the problem of the existence of the Green–Samoilenko function for linear expansions of dynamical systems on a torus of the form $$\frac{{d\phi }}{{dt}} = a(\phi ),{\text{ }}C(\phi )\frac{{d\phi }}{{dt}} + \frac{1}{2}\dot C(\phi )x = A(\phi )x,$$ where C(ϕ) ∈ C′(T m; a) is a nondegenerate symmetric matrix.

Article (Ukrainian)

### On Some Properties of the Behavior of Linear Extensions of Dynamical Systems on a Torus under Perturbation of Phase Variables

Ukr. Mat. Zh. - 2002. - 54, № 3. - pp. 408-412

We investigate classes of linear extensions of dynamical systems on a torus for which the Lyapunov functions exist for an arbitrary flow on the torus. Linear extensions for which the Lyapunov functions exist only with varying coefficients are considered separately. We investigate the problem of preservation of regularity under perturbation of phase variables.