Tkachenko V. I.
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.
Antoniouk A. Vict., Berezansky Yu. M., Boichuk A. 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
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
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
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
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 xn+1 = qxn + fn (xn ,..., xn-k ), n ∈ Z, where nonlinear functions fn satisfy the Yorke condition. For every positive integer k, we represent the interval (0, 1] as the union of [(2k + 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.
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.
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.
Berezansky Yu. M., Boichuk A. 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
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.
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.
Ukr. Mat. Zh. - 1996. - 48, № 1. - pp. 109-115
For a discrete dynamical system ω n =ω0+αn, where a is a constant vector with rationally independent coordinates, on thes-dimensional torus Ω we consider the setL of its linear unitary extensionsx n+1=A(ω0+αn)x n , whereA (Ω) is a continuous function on the torus Ω with values in the space ofm-dimensional unitary matrices. It is proved that linear extensions whose solutions are not almost periodic form a set of the second category inL (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.
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.
Ukr. Mat. Zh. - 1990. - 42, № 10. - pp. 1383–1388
Ukr. Mat. Zh. - 1989. - 41, № 10. - pp. 1379–1383