# Tkachenko V. I.

### Almost periodic solutions of Lotka – Volterra systems with diffusion and impulsive action

Dvornyk A. V., Struk O. O., Tkachenko V. I.

↓ Abstract

Ukr. Mat. Zh. - 2018. - 70, № 2. - pp. 177-192

We establish sufficient conditions for the existence and asymptotic stability of positive piecewise continuous almost periodic solutions for the Lotka –Volterra systems of differential equations with diffusion and impulsive action.

### Anatolii Mykhailovych Samoilenko (on his 80th birthday)

Antoniouk A. Vict., Berezansky Yu. M., Boichuk О. A., Gutlyanskii V. Ya., Khruslov E. Ya., Kochubei A. N., Korolyuk V. S., Kushnir R. M., Lukovsky I. O., Makarov V. L., Marchenko V. O., Nikitin A. G., Parasyuk I. O., Pastur L. A., Perestyuk N. A., Portenko N. I., Ronto M. I., Sharkovsky O. M., Tkachenko V. I., Trofimchuk S. I.

Ukr. Mat. Zh. - 2018. - 70, № 1. - pp. 3-6

### Oleksandr Mykolaiovych Sharkovs’kyi (on his 80th birthday)

Fedorenko V. V., Ivanov А. F., Khusainov D. Ya., Kolyada S. F., Maistrenko Yu. L., Parasyuk I. O., Pelyukh G. P., Romanenko O. Yu., Samoilenko V. G., Shevchuk I. A., Sivak A. G., Tkachenko V. I., Trofimchuk S. I.

Ukr. Mat. Zh. - 2017. - 69, № 2. - pp. 257-260

### Almost periodic solutions of systems with delay and nonfixed times of impulsive action

Dvornyk A. V., Tkachenko V. I.

Ukr. Mat. Zh. - 2016. - 68, № 11. - pp. 1450-1466

We study the existence and asymptotic stability of piecewise continuous almost periodic solutions for systems of differential equations with delay and nonfixed times of impulsive action that can be regarded as mathematical models of neural networks.

### Robustness of exponential dichotomies of boundary-value problems for general first-order hyperbolic systems

Kmit I. Ya., Recke L., Tkachenko V. I.

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.

### On sharp conditions for the global stability of a difference equation satisfying the Yorke condition

Nenya O. I., Tkachenko V. I., Trofimchuk S. I.

Ukr. Mat. Zh. - 2008. - 60, № 1. - pp. 73–80

Continuing our previous investigations, we give simple sufficient conditions for global stability
of the zero solution of the difference equation
*x*_{n+1} = *qx _{n }* +

*f*(

_{n }*x*,...,

_{n }*x*),

_{n-k }*n ∈ Z*, where nonlinear functions

*f*satisfy the Yorke condition. For every positive integer k, we represent the interval (0, 1] as the union of [(2

_{n }*k*+ 2) /3] disjoint subintervals, and, for

*q*from each subinterval, we present a global-stability condition in explicit form. The conditions obtained are sharp for the class of equations satisfying the Yorke condition.

### On Impulsive Lotka–Volterra Systems with Diffusion

Ukr. Mat. Zh. - 2002. - 54, № 4. - pp. 514-526

We study a two-dimensional Lotka–Volterra system with diffusion and impulse action at fixed moments of time. We establish conditions for the permanence of the system. In the case where the coefficients of the system are periodic in *t* and independent of the space variable *x*, we obtain conditions for the existence and uniqueness of periodic solutions of the system.

### On Reducibility of Systems of Linear Differential Equations with Quasiperiodic Skew-Adjoint Matrices

Ukr. Mat. Zh. - 2002. - 54, № 3. - pp. 419-424

We prove that there exists an open set of irreducible systems in the space of systems of linear differential equations with quasiperiodic skew-adjoint matrices and fixed frequency module.

### On linear homogeneous almost periodic systems that satisfy the Favard condition

Ukr. Mat. Zh. - 1998. - 50, № 3. - pp. 409–413

We prove the existence of a linear homogeneous almost periodic system of differential equations that has nontrivial bounded solutions and is such that all systems from a certain neighborhood of it have no nontrivial almost periodic solutions.

### Anatolii Mikhailovich Samoilenko (on his 60th birthday)

Berezansky Yu. M., Boichuk О. A., Korneichuk N. P., Korolyuk V. S., Koshlyakov V. N., Kulik V. L., Luchka A. Y., Mitropolskiy Yu. A., Pelyukh G. P., Perestyuk N. A., Skorokhod A. V., Skrypnik I. V., Tkachenko V. I., Trofimchuk S. I.

Ukr. Mat. Zh. - 1998. - 50, № 1. - pp. 3–4

### On the exponential dichotomy of linear almost periodic pulse systems

Ukr. Mat. Zh. - 1998. - 50, № 1. - pp. 136–142

For a linear almost periodic pulse system, we prove that the exponential dichotomy on a semiaxis implies the exponential dichotomy on the entire axis.

### On uniformly stable linear quasiperiodic systems

Ukr. Mat. Zh. - 1997. - 49, № 7. - pp. 981–987

In a finite-dimensional complex space, we consider a system of linear differential equations with quasiperiodic skew-Hermitian matrix. The space of solutions of this system is a sum of one-dimensional invariant subspaces. Over a torus defined by a quasiperiodic matrix of the system, we investigate the corresponding one-dimensional invariant bundles (nontrivial in the general case). We find conditions under which these bundles are trivial and the system can be reduced to diagonal form by means of the Lyapunov quasiperiodic transformation with a frequency module coinciding with the frequency module of the matrix of the system.

### On the exponential dichotomy of linear difference equations

Ukr. Mat. Zh. - 1996. - 48, № 10. - pp. 1409-1416

We consider a system of linear difference equations*x* ^{n+1} =A (n)x^{n} in an*m*-dimensional real or complex space*Vsum* with det*A(n)* = 0 for some or all*n* ε*Z*. We study the exponential dichotomy of this system and prove that if the sequence {*A(n)*} is Poisson stable or recurrent, then the exponential dichotomy on the semiaxis implies the exponential dichotomy on the entire axis. If the sequence {*A (n)*} is almost periodic and the system has exponential dichotomy on the finite interval {*k*, ...,*k* +*T*},*k* ε*Z*, with sufficiently large*T*, then the system is exponentially dichotomous on*Z*.

### On linear systems with quasiperiodic coefficients and bounded solutions

Ukr. Mat. Zh. - 1996. - 48, № 1. - pp. 109-115

For a discrete dynamical system ω_{ n }=ω_{0}+α*n*, where a is a constant vector with rationally independent coordinates, on the*s*-dimensional torus Ω we consider the set*L* of its linear unitary extensions*x* _{ n+1}=*A*(ω_{0}+α*n*)*x* _{ n }, where*A* (Ω) is a continuous function on the torus Ω with values in the space of*m*-dimensional unitary matrices. It is proved that linear extensions whose solutions are not almost periodic form a set of the second category in*L* (representable as an intersection of countably many everywhere dense open subsets). A similar assertion is true for systems of linear differential equations with quasiperiodic skew-symmetric matrices.

### On the exponential dichotomy of pulse evolution systems

Ukr. Mat. Zh. - 1994. - 46, № 4. - pp. 418–424

The equivalence of regularity and exponential dichotomy is established for linear pulse differential equations with unbounded operators in a Banach space. The separatrix manifolds of a linear pulse system exponentially dichotomous on a semiaxis are studied in a finite-dimensional space. The conditions of weak regularity of this system are given.

### On linear almost periodic pulse systems

Ukr. Mat. Zh. - 1993. - 45, № 1. - pp. 105–113

For a linear almost periodic system under pulse influence, the conditions are established under which this system is reducible (by a linear change of variables with a discontinuous almost periodic matrix) to a system without pulses but with a Bohr almost periodic right-hand side. The set of linear almost periodic pulse systems possessing only bounded solutions is studied.

### Splitting and the spectrum of a linear differential equation with quasiperiodic coefficients

Ukr. Mat. Zh. - 1990. - 42, № 10. - pp. 1383–1388

### The green function and conditions for the existence of invariant sets of impulse systems

Ukr. Mat. Zh. - 1989. - 41, № 10. - pp. 1379–1383