On the dichotomy of a system of linear differential equations with conditionally periodic coefficients

We show that a system of linear differential equations with conditionally periodic coefficients is exponentially dichotomous if and only if the spectrum of the monodromy operator does not meet the unit circle.     

Differential equations with almost periodic or conditionally periodic coefficients: Recurrence and reducibility

Recurrence of an individual trajectory of a linear nonautonomous differential equation on a compact Lie group for the case of an almost periodic or conditionally periodic dependence of the right-hand side on time is proved. The relation between recurrence and reducibility is examined.Translated fromMatematicheskie Zametki, Vol. 64, No. 2, pp. 229–237, August, 1998.This research was supported by the Russian Foundation for Basic Research under grants No. 96-01-00378 and No. 96-15-96072.     

Oscillation theorem for higher-order linear differential equations with periodic coefficients

In this paper, the zeros of solutions for higher-order linear differential equations with periodic coefficients are studied. It is shown that under certain hypotheses, the convergence exponent of zeros of the product of every fundamental solution is infinite.     

PERTURBATION RESULTS OF SECOND ORDER LINEAR DIFFERENTIAL EQUATIONS WITH PERIODIC COEFFICIENTS

Two perturbation results on the linear differential function f" ∏(z)A(z)f = 0 areobtained, where ∏(z) and A(z) are periodic entire functions with period 2πi and σe(∏) ＜σe(A).     

Reducibility of linear systems of difference equations with almost periodic coefficients

We establish sufficient conditions of the reducibility of the linear system of difference equationsx(t+1)=Ax(t) + P(t) x(t) with an almost periodic matrixP(t) to a system with a constant matrix.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 45, No. 12, pp. 1661–1667, December, 1993.     

Exponential dichotomy of systems of linear difference equations with periodic coefficients

We study the problem of exponential dichotomy for the systems of linear difference equations with periodic coefficients. Some criterion is established for exponential dichotomy in terms of solvability of a special boundary value problem for a system of discrete Lyapunov equations. We also give estimates for dichotomy parameters.     

Systems of differential equations with periodic coefficients

We consider linear systems of differential equations with periodic coefficients. We prove the solvability of nonhomogeneous systems in the Sobolev space W ₂ ¹ (R) and establish estimates for the solutions. This result implies a perturbation theorem for the exponential dichotomy of systems of differential equations with periodic coefficients.     

Stability criterion for second order linear impulsive differential equations with periodic coefficients

In this paper we obtain instability and stability criteria for second order linear impulsive differential equations with periodic coefficients. Further, a Lyapunov type inequality is also established. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

The loss of the property of reducibility by systems of linear differential equations with almost periodic coefficients

It is proved that among systems with almost periodic analytic coefficients (with an algebraic number as frequency base), there are systems which lose the property of reducibility on the boundary of some disk of values of the parameter.Translated from Matematicheskie Zametki, Vol. 8, No. 1, pp. 115–120, July, 1970.In conclusion the author wishes to express his gratitude to L. B. Danilov and M. G. Rabinovich for their comments on this work and also to D. V. Anosov for his suggestions.     

Reducibility of a linear system of differential equations with odd almost periodic coefficients

A theorem on the Kolmogorov reducibility of a system of ordinary differential equations with odd almost periodic coefficients is proved. Translated fromMatematicheskie Zametki, Vol. 60, No. 1, pp. 109–119, July, 1996.     

Stability of solutions to delay differential equations with periodic coefficients of linear terms

Under study are the systems of quasilinear delay differential equations with periodic coefficients of linear terms. We establish sufficient conditions for the asymptotic stability of the zero solution, obtain estimates for solutions which characterize the decay rate at infinity, and find the attractor of the zero solution. Similar results are obtained for systems with parameters.     

On periodic solutions of countable systems of linear and quasilinear difference equations with periodic coefficients

We present conditions for the existence of periodic solutions of linear difference equations with periodic coefficients in spaces of bounded number sequences. In the case where the generating linear equation has a unique periodic solution, we indicate sufficient conditions for the existence of a periodic solution of a quasilinear difference equation.