Integral and Differential Equations Department

Scientific activity

Scientific scool Doctors of science Main research interests Main results Monographs

Kyiv School of nonlinear mechanics, differential equations and mathematical physics, founded by M. M. Bogolyubov, are functioning for more than 60 years. The glory of this School were developed and multiplied by the world-renowned scientists: academician of NAS of Ukraine and RAS Yu.O. Mytropolsky, academicians of NAS of Ukraine O.S. Parasyuk, A.M. Samoilenko, O.M. Sharkovsky, D.Y. Petryna, M.O. Perestyuk, corresponding member of NAS of Ukraine V.I. Fuschych. To the list of significant achievements of the School in differential equations can be added: development of method of Green function in invariant sets theory regarding its applications to equations with delay, difference, impulsive, singularly perturbed and stochastic equations (professors A.M. Samoilenko, D.I. Martynyuk, M.O.Perestyuk, O.M. Stanzhytsky, docents V.V. Ischuk, M.Y.Svishchuk, V.Y. Danilov, G.V. Veriovkina); development of theory of impulsive systems (professors A.M. Samoilenko, M.O. Perestyuk, V. G. Samoilenko, docent O.C. Chernikova, research ass. Y.I. Samoilenko); development of methods of constructing of asymptotic representations of solutions of non-autonomous systems of differential equations (docent V. M. Burym); development of theory of coisotropic invariant tori of Hamiltonian and locally Hamiltonian systems (professor I.O.Parasyuk, Y.V.Loveikin); development of methods of investigation of stability and exponential dichotomy of stochastic systems (professors A.M. Samoilenko, O.M. Stanzhytsky); development of abstract theory of multivalued semi-processes and its global attractors (professors M.O. Perestyuk, O.V. Kapustyan) development of differential-operator inclusions and variational inequalities (P.O. Kasyanov, N.V. Zadoyanchuk); optimal control (professor O.V. Kapustyan, A.V. Sukretna).

Doctors of science

1982: D.I.Martinuk,

1986: M.O.Perestyuk, M.I.Ronto,

1988: V.L.Kulik,

1992: O.A.Boichuk, M.I.Ilolov, Yu.V.Teplinskiy,

1993: V.P.Yakovec,

1994: M.U.Ahmetov, S.I.Trofimchuk,

1995: K.Kenzhebaev, I.O.Parasyuk, V.V.Marinec, R.I.Petrishin,

1996: V.B.Moseenkov,

1997: V.I.Tkachenko,

1998: M.M.Pritula, S.D.Borisenko,

2002: O.M.Stanzhickiy,

2003: V.I.Lagno,

2004: M.F.Gorodniy, I.M.Cherevko, L.I.Karandzhulov,

2005: A.M.Ronto,

2006: Ya.A.Prikarpatskiy,

2007: E.P.Belan,

2009: O.V.Kapustyan, O.Yu.Shvec, Ya.I.Bigun,

2010: I.I.Korol, P.O.Kasyanov,

2012: A.V.Chaikovskii,

2014: I.M.Hrod,

2015: I.I.Klevchuk,

2016: N.V.Skrypnyk,

2019: R.M.Taranec.

Main research interests

Theory of integral and invariant sets of nonlinear dynamic systems.

Asymptotic methods and methods of averaging of nonlinear mechanics.

Theory of impulsive systems.

Numerical-analytical methods for solving boundary value problems.

Multi-frequency oscillations in Hamiltonian systems.

Study of asymptotic behavior of solutions of differential equations.

Stability theory of stochastic differential equations.

Investigation of the qualitative behavior of solutions of multivalued evolution systems.

The theory of optimal control.

Main results

– Development of analytical theory of differential equations, study the existence of solutions of linear differential equations with polynomial coefficients (K.Ya.Latysheva, I.A.Pavlyuk, M.I.Tereschenko).

– Analysis of asymptotic properties of solutions of nonautonomous systems of differential equations (I.A.Pavlyuk, V.M.Burym).

– Development of methods of asymptotic theory of nonlinear oscillations (Yu.O.Mytropolskyy, B.I.Moseyenkov).

– Development of Green's function method in the theory of invariant sets and on this basis, prooving theorems of existence of invariant manifolds for equations with delay, difference, impulsive and stochastic equations (A.M.Samoilenko, D.I.Martynyuk, M.O.Perestyuk, V. V.Ischuk, O.M.Stanzhytskyy, V.Ya.Danilov).

– Creation of the theory of systems with impulse action. Development methods for investigating stability, the study of periodic and almost periodic solutions for such systems (A.M.Samoilenko, M.O.Perestyuk, M.U.Ahmetov, O.S.Chernikova).

– Development of numerical-analytical method of investigation of periodic solutions of essentially nonlinear systems, systems with delay and impulsive systems (A.M.Samoilenko, M.Y.Ronto, M.O.Perestyuk, D.I.Martynyuk, K.Kenzhebayev).

– Development of perturbation theory of coisotropic invariant tori of Hamiltonian and locally Hamiltonian Systems (I.O.Parasyuk, Yu.V.Loveikin).

– Development of methods for the study of stability of stochastic systems (A.M.Samoilenko, O.M.Stanzhytskyy, M.Ya.Svishchuk).

– Development of the methods of the theory of global attractors and qualitative study of evolution systems without uniqueness, differential-operator inclusions and variational inequalities (V.S. Melnik, O.V.Kapustyan, P.O.Kasyanov, N.V.Zadoyanchuk).

– Research on optimal control of systems with distributed parameters, stochastic and impulsively perturbed systems (M.O.Perestyuk, O.M.Stanzhytskyy, O.V.Kapustyan, A.V.Sukretna).

Monographs

Mitropol'skii Y.A., Martynyuk D.I. Lectures on the theory of oscillations of systems with delay. - Kiev: Institute of Mathematics, 1969.

Latisheva K.Y., Tereshchenko M.І. Lectures on the analytic theory of differential equations and their applications. - Kiev: Vyshcha Schkola, 1970.

Pavlyuk I.A. Asymptotic properties of solutions of nonautonomous systems of differential equations of second order. - Kiev: Vyscha Schkola, 1970.

Mitropol'skii Y.A., Martynyuk D.I. Lectures on the qualitative theory of difference equations. - Kiev: Institute of Mathematics, 1972.

Mitropol'skii Y.A., Moseenkov B.I.Asymptotic solutions of partial differential equations. - Kiev: Vyshcha Schkola, 1976.

Mitropol'skii Y.A., Martynyuk D.I. Periodic and quasi-periodic oscillations of systems with delay. - Kiev: Vyshcha Schkola, 1979.

Mitropol'skii Yu.A., Samoilenko A.M., Martynyuk D.I. System of evolution equations with periodic and conditionally periodic coefficients. - Kiev: Naukova Dumka, 1984.

Samoilenko A.M. Elements of the mathematical theory of multi-frequency oscillations: invariant tori. - Moscow: Nauka, 1987.

Samoilenko A.M., Perestyuk N.A. Differential equations with impulse action. - Kiev: Vyshcha School, 1987.

Mitropol'skii Yu.A., Samoilenko A.M. Mathematical problems of nonlinear mechanics. - Kiev: Vyshcha School, 1987.

Samoilenko A.M. Elements of the Mathematical Theory of Multi-Frequency Oscilations. (Mathematics and its Applications, V. 71). – Dortrecht:Kluwer Academic Publishers Group, 1991.

Martynyuk D.I.,Mitropolsky Yu.A., Samoilenko A.M. Systems of evolution equations with periodic and quasiperiodic coefficients. – Dordrecht – Boston – London: Kluwer Acad. Publ., 1992.

Samoilenko A.M., Perestyuk N.A. Impulsive differential equations. – World Scientific, 1995.

Gonchar M.S. Stock market and economic growth. - Kyiv: Oberegy, 2001.

Perestyuk M.O., Bobochko V.N. Asymptotic integration of the Liouville equation with turning points. - Kyiv: Naukova Dumka, 2002.

Perestyuk N.A., Plotnikov V.A., Samoilenko A.M., Skrypnyk N.V. Impulse differential equations with discontinuous and multivalued right-hand side. - Kiev: Institute of Mathematics, 2007.

Kapustyan O.V., Mel'nik V.S., Valero J., Yasinsky V.V. Global attractors of multi-valued dynamical systems and evolution equations without uniquness. – Kyiv: Naukova dumka, 2008.

Zgurovsky M.Z., Kasyanov P.O., Melnik V.S. The differential-operator inclusions and variational inequalities in infinite-dimensional spaces. - Kiev: Naukova Dumka, 2008.

Samoilenko A.M., Stanzhytskiy O.M. Qualitative and asymptotic analysis of differential equations with random perturbations. - Kyiv: Naukova Dumka, 2009.