Nonlinear dynamical systems, KAM-theory, invariant manifolds, singular boundary value problems

Previous and Current Research

We are focused on the theory of multifrequency nonlinear oscillations and invariant manifolds, development of methods for analyzing Lagrangian and Hamiltonian systems on manifolds (in particular, KAM-theory, variational methods for detecting quasiperiodic solutions), qualitative methods for studying nonlinear singular BVP for ordinary differential equations on semi-axis, dynamic bifurcations in fast-slow systems, the theory of bounded and quasi-periodic solutions of indefinite monotone systems on Riemannian manifolds.

We have priority in proving theorems on the conservation of multifrequency quasi-periodic motions under small perturbations of integrable reversible systems and locally Hamiltonian systems in the most complex case where the number of so-called slow variables is significantly less than the dimension of the frequency basis. We described new phenomena accompanying the simultaneous deformation of both the Hamiltonian function and the symplectic structure of a completely integrable Hamiltonian system. We dicovered Hamiltonian systems with non-exact symplectic structure that have compact invariant nilmanifolds carrying nilpotent flows.

Future Projects and Goals

• Studying periodic and quaiperiodic motions of quaternionic dynamical systems.

• Theorems on the existence and bifurcation of invariant sections for dynamical systems on fiber bundles.

 Collaboration

• Institute of Mathematics NAS of Ukraine

•  Austro-Ukrainian Institute for Science and Technology (AUI), TU Wien

Methodological and Technical Expertise

Nonlinear dynamics, small denominators problems.

The team:

Mathematicians

DSc, Professor M. Horodniy

DSc, Professor O. Pryshlyak

Phd, Associate Professor V. Zhuravlev

Phd, Associate Professor V. Babych

Phd, Assistant Professor S. Bilun

Phd, Assistant Professor I. Tsyganivska

Phd Student A. Prus

Phd Student V. Kravets

Selected Publications

1.     Parasyuk, Igor. Landau–Kolmogorov type inequalities for curves on Riemannian manifolds / Igor Parasyuk // Mathematical Inequalities and Applications. 2019. Vol. 22, no. 2. P. 433–443.

2.     Parasyuk, I. O. Quasiperiodic Forced Oscillations of a Solid Body in the Field of a Quadratic Potential / I. O. Parasyuk // Journal of Mathematical Sciences. 2019. Vol. 240, no. 3. P. 323–341.

3.     Parasyuk, Igor O. Hyperbolic quasiperiodic solutions of U-monotone systems on Riemannian manifolds / Igor O. Parasyuk // Dynamics of Continuous, Discrete and Impulsive Systems, Series A: Mathematical Analysis. 2019. Vol. 26, no. 1. P. 21–52.

4.     Parasyuk, I. O., Repeta, B. V. Hyperbolic Invariant Tori of a Fast-Slow System with Dynamic Bifurcation of Multifrequency Oscillations// Journal of Mathematical Sciences. 2017. 222. P. 312-335.

5.     Parasyuk, Igor. Dynamical bifurcation in a system of coupled oscillators with slowly varying parameters / Igor Parasyuk, Bogdan Repeta // Electronic Journal of Differential Equations. 2016. Vol. 2016, no. 233. P. 1–32.

6.     Parasyuk, I. Quasiperiodic Extremals of Nonautonomous Lagrangian Systems on Riemannian Manifolds. / I. Parasyuk // Ukrainian Mathematical Journal. 2015. Vol. 66, no. 10. P. 1387-1406.

7.     Samoilenko, A. M. Dynamical bifurcation of multifrequency oscillations in a fast-slow system / A. M. Samoilenko, I. O. Parasyuk, B. V. Repeta // Ukrainian Mathematical Journal. 2015. Vol. 67, no. 7. P. 1008–1037.

8.     Lahoda, V. Theorem on the existence of an invariant section over Rm for the indefinite monotone system in Rn. / V. Lahoda, I. Parasyuk // Ukrainian Mathematical Journal. 2013. Vol. 65, no. 1.

9.     Lagoda, V. A. Lipschitzian invariant tori of indefinite monotone system / V. A. Lagoda, I. O. Parasyuk, A. M. Samoilenko // Ukrainian Mathematical Journal. 2012. Vol. 64, no. 3. P. 408–432.

10. Parasyuk, Igor. Variational approach for weak quasiperiodic solutions of quasiperiodically excited Lagrangian systems on Riemannian manifolds / Igor Parasyuk, Anna Rustamova // Electronic Journal of Differential Equations. 2012. Vol. 2012, no. 66. P. 1–22.

(Scopus)Parasyuk, Igor O. Author ID: 16430162800

Contacts

parasyuk@knu.ua

Postal address: Taras Shevchenko National University of Kyiv, Faculty of Mechanics and Mathematics, Ukraine, Kyiv-01601, Glushkov Av. 4e