# Lykova O. B.

### Amplitude synchronization in a system of two coupled semiconductor lasers

Lykova O. B., Schneider K. R., Yanchuk S. V.

Ukr. Mat. Zh. - 2008. - 60, № 3. - pp. 426–435

We consider a system of ordinary differential equations used to describe the dynamics of two coupled single-mode semiconductor lasers. In particular, we study solutions corresponding to the amplitude synchronization. It is shown that the set of these solutions forms a three-dimensional invariant manifold in the phase space. We study the stability of trajectories on this manifold both in the tangential direction and in the transverse direction. We establish conditions for the existence of globally asymptotically stable solutions of equations on the manifold synchronized with respect to the amplitude.

### Olexiy Bogolyubov (03.25.1911 - 01.11.2004)

Dobrovol'skii V. A., Lykova O. B., Mitropolskiy Yu. A., Pustovoytov M. O., Samoilenko A. M., Urbansky V. M.

Ukr. Mat. Zh. - 2006. - 58, № 4. - pp. 564–567

### Central manifolds of quasilinear parabolic equations

Ukr. Mat. Zh. - 1998. - 50, № 3. - pp. 315–328

We investigate central manifolds of quasilinear parabolic equations of arbitrary order in an unbounded domain. We suggest an algorithm for the construction of an approximate central manifold in the form of asymptotically convergent power series. We describe the application of the results obtained in the theory of stability.

### On properties of central manifolds of a stationary point

Ukr. Mat. Zh. - 1997. - 49, № 1. - pp. 68–76

We present results concerning properties of central manifolds of a stationary point. The results are illustrated by examples.

### Theorem on the central manifold of a nonlinear parabolic equation

Ukr. Mat. Zh. - 1996. - 48, № 8. - pp. 1021-1036

Under certain assumptions, we prove the existence of an *m*-parameter family of solutions that form the central invariant manifold of a nonlinear parabolic equation. For this purpose, we use an abstract scheme that corresponds to energy methods for strongly parabolic equations of arbitrary order.

### Integral manifolds and exponential splitting of linear parabolic equations with rapidly varying coefficients

Ukr. Mat. Zh. - 1995. - 47, № 12. - pp. 1593–1608

We study linear parabolic equations with rapidly varying coefficients. It is assumed that the averaged equation corresponding to the source equation admits exponential splitting. We establish conditions under which the source equation also admits exponential splitting. It is shown that integral manifolds play an important role in constructing transformations that split the equations under consideration. To prove the existence of integral manifolds, we apply Zhikov's results on the justification of the averaging method for linear parabolic equations.

### On the reduction principle in the theory of stability of motion

Ukr. Mat. Zh. - 1993. - 45, № 12. - pp. 1653–1660

This paper deals with the development of Lyapunov's idea of reducing the problem of stability of the trivial solution of a system of higher-order differential equations to a similar problem for a system of lower order. Special attention is paid to the application of integral manifolds and approximate integral manifolds.

### Integral manifolds and the reduction principle in stability theory. IV

Ukr. Mat. Zh. - 1991. - 43, № 12. - pp. 1696–1702

### Integral manifolds and the reduction principle in stability theory. III

Ukr. Mat. Zh. - 1991. - 43, № 10. - pp. 1324–1329

### Integral manifolds and a reduction principle in stability theory. II

Ukr. Mat. Zh. - 1990. - 42, № 10. - pp. 1315–1321

### Integral manifolds and a reduction principle in stability theory

Ukr. Mat. Zh. - 1989. - 41, № 12. - pp. 1607–1613

### Asymptotic expansions of invariant manifolds. III

Ukr. Mat. Zh. - 1989. - 41, № 8. - pp. 1033–1041

### Asymptotic expansions of invariant manifolds. II

Ukr. Mat. Zh. - 1988. - 40, № 6. - pp. 709-716

### Construction of periodic solutions of nonlinear systems in critical cases

Ukr. Mat. Zh. - 1988. - 40, № 1. - pp. 62-69

### Asymptotic expansions of invariant manifolds. I

Ukr. Mat. Zh. - 1987. - 39, № 4. - pp. 411–418

### Application of sign-constant functions to the theory of integral manifolds

Lykova O. B., Vladimirov V. N.

Ukr. Mat. Zh. - 1987. - 39, № 2. - pp. 190-194

### The development of methods of nonlinear mechanics in the works of Yu. A. Mitropol'skii

Lykova O. B., Samoilenko A. M.

Ukr. Mat. Zh. - 1987. - 39, № 1. - pp. 534–538

### Solutions of systems of differential equations, bounded with respect to part of the variables

Ukr. Mat. Zh. - 1985. - 37, № 2. - pp. 139 – 146

### Problem of the existence of integral manifolds

Ukr. Mat. Zh. - 1983. - 35, № 1. - pp. 1—8

### On the contraction principle for a differential equation with unbounded operator coefficient

Ukr. Mat. Zh. - 1975. - 27, № 2. - pp. 240–243

### Construction of a lyapunov functional for a weakly nonautonomic linear equation in Hilbert space

Ukr. Mat. Zh. - 1974. - 26, № 1. - pp. 90–95

### Method of constructing Lyapunov's function for weakly nonautonomous linear systems of differential equations

Ukr. Mat. Zh. - 1972. - 24, № 5. - pp. 634–641

### The reduction principle in Banach space

Ukr. Mat. Zh. - 1971. - 23, № 4. - pp. 464–471

### The reductibility of some differential equations in Banach space

Ukr. Mat. Zh. - 1968. - 20, № 5. - pp. 628–641

### Investigation of a class of nonlinear differential equations in Hilbert space

Ukr. Mat. Zh. - 1967. - 19, № 3. - pp. 112–117

### Yurii Alekseevich Mitropol'skii (on his 50th birthday)

Glushkov V. M., Korolyuk V. S., Lykova O. B., Parasyuk O. S.

Ukr. Mat. Zh. - 1967. - 19, № 1. - pp. 3–8

### On the integral manifold of a nonlinear system in a Hilbert space

Lykova O. B., Mitropolskiy Yu. A.

Ukr. Mat. Zh. - 1965. - 17, № 5. - pp. 43-53

### On an integral manifold of nonlinear differential equations containing slow and fast motions

Lykova O. B., Mitropolskiy Yu. A.

Ukr. Mat. Zh. - 1964. - 16, № 2. - pp. 157-163

The authors establish the existence and properties of an $s + 1$ -dimensional local integral manifold of a system of $l + m + n$ nonlinear differential equations of the form $$\frac{dx}{dt} = X(y,z)x + \varepsilon X_1(t, x, y, z),$$ $$\frac{dy}{dt} =Y(x, z), y + \varepsilon Y_1 (t, x, y, z),$$ $$\frac{dz}{dt} = \varepsilon Z_1 (t, x, y, z),$$ where $x, y$ characterize the fast, and $z$ the slow motions.

### Investigation of the solutions of a system of $n + m$ nonlineai differential equations in the vicinity of an integral manifold

Ukr. Mat. Zh. - 1964. - 16, № 1. - pp. 13-30

For a system of $n + m$ equations $$\frac{dx}{dt} = X(y)x + \varepsilon X*(t, x, y),$$ $$\frac{dy}{dt} = \varepsilon Y(t, x, y),$$ where $x, X*, y, Y$ are respectively $n$ and $m$ vectors, $X — n \times n$ is the matrix, $\varepsilon$ is a small parameter, the author proves the theorem of the existence and properties of a two-dimensional local integral manifold in the neighbourhood of family of periodic solutions $$x = 0,\; y = y^0(\psi, a)$$ oi the lollowing auxiliary system $$\frac{dx}{dt} = X(y)x,$$ $$\frac{dy}{dt} = \varepsilon Y_0(x, y),$$ where $$Y_0(x, y) = \lim_{T\rightarrow 0}\int_0^T Y(t, x,y)dt.$$

### On Periodic Solutions of Systems of Nonlinear Equations with a Small Parameter

Lykova O. B., Mitropolskiy Yu. A.

Ukr. Mat. Zh. - 1960. - 12, № 4. - pp. 391 - 401

The authors consider a system of nonlinear differential equations containing a small parameter with undifferentiated right parts of types (1) and (37). Making some assumptions, the existence unique and asymptotic stability of a periodic solution is proved for such systems, and an estimate is found for the difference between the exact solution of the systems under consideration and their first approximation, which can be found without any essential difficulty.

### On Certain Properties of the Solutions of Systems of Nonlinear Differential Equations with Slowly Varying Parameters

Ukr. Mat. Zh. - 1960. - 12, № 3. - pp. 267 - 278

In a previous paper the author proposed an algorithm for finding an approximate (with precision up to a magnitude of order e) two-parameter family of special solutions of the system. In this paper the existence and uniqueness ot a corresponding exact two-parameter family of solutions of system (1) is proved; the difference between the exact family of solutions and its mth approximation is shown to be of the order of em; the property of attraction to the found approximate family of solutions is established for any solutions of system (1) having initial values which belong to the region of definition of the exact two-parameter family of solutions of system (1)