# Parasyuk I. O.

### 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

### On the 100th birthday of outstanding mathematician and mechanic Yurii Oleksiiovych Mytropol’s’kyi (03.01.1917 – 14.06.2008)

Berezansky Yu. M., Boichuk О. A., Korolyuk V. S., Lukovsky I. O., Makarov V. L., Nikitin A. G., Parasyuk I. O., Perestyuk N. A., Samoilenko A. M., Sharkovsky O. M.

Ukr. Mat. Zh. - 2017. - 69, № 1. - pp. 132-144

### Mykola Oleksiiovych Perestyuk (on his 70th birthday)

Boichuk О. A., Gorbachuk M. L., Gorodnii M. F., Khruslov E. Ya., Lukovsky I. O., Makarov V. L., Parasyuk I. O., Samoilenko A. M., Samoilenko V. G., Sharkovsky O. M., Shevchuk I. A., Slyusarchuk V. Yu., Stanzhitskii A. N.

Ukr. Mat. Zh. - 2016. - 68, № 1. - pp. 142-144

### Dynamical Bifurcation of Multifrequency Oscillations in a Fast-Slow System

Parasyuk I. O., Repeta B. V., Samoilenko A. M.

Ukr. Mat. Zh. - 2015. - 67, № 7. - pp. 890-915

We study a dynamical analog of bifurcations of invariant tori for a system of interconnected fast phase variables and slowly varying parameters. It is shown that, in this system, due to the slow evolution of the parameters, we observe the appearance of transient processes (from the damping process to multifrequency oscillations) asymptotically close to motions on the invariant torus.

### Quasiperiodic Extremals of Nonautonomous Lagrangian Systems on Riemannian Manifolds

Ukr. Mat. Zh. - 2014. - 66, № 10. - pp. 1387–1406

The paper deals with a quasiperiodically excited natural Lagrangian system on a Riemannian manifold. We find sufficient conditions under which this system has a weak Besicovitch quasiperiodic solution minimizing the averaged Lagrangian. It is proved that this solution is indeed a twice continuously differentiable uniformly quasiperiodic function, and the corresponding system in variations is exponentially dichotomous on the real axis.

### Theorem on the existence of an invariant section over $\mathbb{R}^m$ for the indefinite monotone system in $\mathbb{R}^m \times \mathbb{R}^n$

Ukr. Mat. Zh. - 2013. - 65, № 1. - pp. 103-118

We consider a nonlinear system on the direct product $\mathbb{R}^m \times \mathbb{R}^n$. For this system, under the conditions of indefinite coercivity and indefinite monotonicity, we establish the existence of a bounded Lipschitzian invariant section over $\mathbb{R}^m$.

### Estimation of the number of ultrasubharmonics for a two-dimensional almost autonomous Hamiltonian system periodic in time

Ukr. Mat. Zh. - 2012. - 64, № 4. - pp. 463-489

Using the Arnold method of detection of fixed points of symplectic diffeomorphisms, we find lower estimates for the number of ultrasubharmonics in a Hamiltonian system on a two-dimensional symplectic manifold with almost autonomous time-periodic Hamiltonian. We show that the asymptotic behavior of these estimates as the perturbation parameter tends to zero depends on which of the four zones of a ring domain foliated by closed level curves of the unperturbed Hamiltonian the generating unperturbed ultrasubharmonics belong to.

### Lipschitzian invariant tori of indefinite monotone system

Lagoda V. A., Parasyuk I. O., Samoilenko A. M.

Ukr. Mat. Zh. - 2012. - 64, № 3. - pp. 363-383

We consider a nonlinear system in the direct product of a torus and a Euclidean space. For this system, under the conditions of indefinite coercivity and indefinite monotonicity, we establish the existence of a Lipschitzian invariant section.

### Invariant tori of locally Hamiltonian systems close to conditionally integrable systems

Loveikin Yu. V., Parasyuk I. O.

Ukr. Mat. Zh. - 2007. - 59, № 1. - pp. 71–98

We study the problem of perturbations of quasiperiodic motions in the class of locally Hamiltonian systems. By using methods of the KAM-theory, we prove a theorem on the existence of invariant tori of locally Hamiltonian systems close to conditionally integrable systems. On the basis of this theorem, we investigate the bifurcation of a Cantor set of invariant tori in the case where a Liouville-integrable system is perturbed by a locally Hamiltonian vector field and, simultaneously, the symplectic structure of the phase space is deformed.

### Nikolai Perestyuk (60th birthday)

Mitropolskiy Yu. A., Parasyuk I. O., Samoilenko A. M., ShkiI N. I.

Ukr. Mat. Zh. - 2006. - 58, № 1. - pp. 113-114

### V. G. Georgii Mykolaiovych Polozhyi (on his 90th birthday)

Glushchenko A. A., Lyashko I. I., Mitropolskiy Yu. A., Parasyuk I. O., Samoilenko A. M., Samoilenko V. G.

Ukr. Mat. Zh. - 2004. - 56, № 4. - pp. 560-561

### Construction of Floquet–Bloch Solutions and Estimation of Lengths of Resonance Zones of One-Dimensional Schrödinger Equation with Smooth Potential

Denysenko O. M., Parasyuk I. O.

Ukr. Mat. Zh. - 2004. - 56, № 1. - pp. 3-18

A one-dimensional Schrödinger equation with a potential characterized by a certain rate of approximation by trigonometric polynomials is investigated by methods of the KAM theory. Estimates for resonance energy zones are obtained. The case where the potential belongs to the Gevrey class is analyzed.

### Bifurcation of a Whitney-Smooth Family of Coisotropic Invariant Tori of a Hamiltonian System under Small Deformations of a Symplectic Structure

Kubichka A. A., Parasyuk I. O.

Ukr. Mat. Zh. - 2001. - 53, № 5. - pp. 610-624

We investigate the influence of small deformations of a symplectic structure and perturbations of the Hamiltonian on the behavior of a completely integrable Hamiltonian system. We show that a Whitney-smooth family of coisotropic invariant tori of the perturbed system emerges in the neighborhood of a certain submanifold of the phase space.

### Generalized and classical almost periodic solutions of Lagrangian systems convex on a compact set

Parasyuk I. O., Zakharin S. F.

Ukr. Mat. Zh. - 1998. - 50, № 12. - pp. 1601–1608

By using the variational method, we establish sufficient conditions for the existence of generalized Besicovitch almost (quasi)periodic solutions and classical quasiperiodic solutions of natural Lagrangian systems with force functions convex on a compact set.

### Perturbations of degenerate coisotropic invariant tori of Hamiltonian systems

Ukr. Mat. Zh. - 1998. - 50, № 1. - pp. 72–86

We consider a Hamiltonian system with a one-parameter family of degenerate coisotropic invariant tori. We prove a theorem on the preservation of the majority of tori under small perturbations of the Hamiltonian.

### Nilpotent flows of S^{1}-invariant Hamiltonian systems on 4-dimensional symplectic manifolds

Parasyuk I. O., Samoilenko A. M.

Ukr. Mat. Zh. - 1997. - 49, № 1. - pp. 122–140

We investigate S^{1}-invariant Hamiltonian systems on compact 4-dimensional symplectic manifolds with free symplectic action of a circle. We show that, in a rather general case, such systems generate ergodic flows of types (quasiperiodic and nilpotent) on their isoenergetic surfaces. We solve the problem of straightening of these flows.

### On the integral of a function along the trajectories of a nilpotent flow

Parasyuk I. O., Samoilenko A. M.

Ukr. Mat. Zh. - 1995. - 47, № 6. - pp. 837–847

We establish conditions under which the integral of a function along a nilpotent flow on the Heisenberg-Iwasawa manifold increases not faster than $|t|^{1/2+ ε},\; 0 < ε < 1$ and indicate cases where this integral can be represented as a superposition of a function defined on a nilmanifold and a nilpotent flow.

### Reduction and coisotropic invariant tori of Hamiltonian systems with non-poisson commutative symmetries. II

Ukr. Mat. Zh. - 1994. - 46, № 7. - pp. 904–914

Haamiltonian systems of mechanical type are considered on a twisted cotangent stratification of a manifold admitting a smooth free torus action. In the case where these systems possess non-Poisson symmetries generated by the torus action, the Lee-Cartan reduction scheme is described and the structure of a reduced phase space and reduced Hamiltonians is clarified.

### Reduction and coisotropic invariant tori of Hamiltonian systems with non-poisson commutative symmetries. I

Ukr. Mat. Zh. - 1994. - 46, № 5. - pp. 537–544

Hamiltonian systems invariant under the non-Poisson torus action are studied on a symplectic manifold. Conditions are established under which coisotropic invariant tori filled with quasiperiodic motions exist in these systems.

### Variables of the action-angle type on symplectic manifolds stratified by coisotropic tori

Ukr. Mat. Zh. - 1993. - 45, № 1. - pp. 77–85

A symplectic manifold is considered under the assumption that a smooth symplectic action of a commutative Lie group with compact coisotropic orbits is defined on it. The problem of existence of variables of the action-angle type is investigated with a view to giving a detailed description of flows in Hamiltonian systems with invariant Hamiltonians. We introduce the notion of a nonresonance symplectic structure for which the problem of recognition of resonance and nonresonance tori is solved.

### Qualitative analysis of families of bounded solutions of the multidimensional nonlinear Schrodinger equation

Fushchich V. I., Parasyuk I. O.

Ukr. Mat. Zh. - 1991. - 43, № 6. - pp. 821-829

### Qualitative analysis of families of bounded solutions of the nonlinear three-dimensional schrodinger equation

Fushchich V. I., Parasyuk I. O.

Ukr. Mat. Zh. - 1990. - 42, № 10. - pp. 1344–1349

### Coisotropic invariant tori of hamiltonian systems of the quasiclassical theory of motion of a conduction electron

Ukr. Mat. Zh. - 1990. - 42, № 3. - pp. 346-351

### Conservation of multidimensional invariant tori of Hamiltonian systems

Ukr. Mat. Zh. - 1984. - 36, № 4. - pp. 467 – 473

### Parameteic resonance in linear Hamiltonian systems with quasiperiodic coefficients

Ukr. Mat. Zh. - 1979. - 31, № 1. - pp. 92–95

### Zones of instability of the Schrödinger equation with a smooth quasiperiodic potential

Ukr. Mat. Zh. - 1978. - 30, № 1. - pp. 70–78

### The application of the Wiener-Levinson theory to the investigation of the asymptotic behavior of continuous bounded solutions of integral equations with difference kernels

Ukr. Mat. Zh. - 1974. - 26, № 3. - pp. 408–411

### On the asymptotic behavior of unbounded solutions of a system of integral equations with a difference kernel

Ukr. Mat. Zh. - 1974. - 26, № 2. - pp. 256–259