Albert Morozov
pr. Gagarina 23, Nizhny Novgorod, 603950 Russia
N.I. Lobachevsky State University of Nizhny Novgorod
Publications:
Morozov K. E., Morozov A. D.
QuasiPeriodic Parametric Perturbations of TwoDimensional Hamiltonian Systems with Nonmonotonic Rotation
2024, vol. 29, no. 1, pp. 6577
Abstract
We study nonconservative quasiperiodic (with $m$ frequencies) perturbations of
twodimensional Hamiltonian systems with nonmonotonic rotation. It is assumed that the
perturbation contains the socalled parametric terms. The behavior of solutions in the vicinity
of degenerate resonances is described. Conditions for the existence of resonance $(m + 1)$dimensional invariant tori for which there are no generating ones in the unperturbed system
are found. The class of perturbations for which such tori can exist is indicated. The results are
applied to the asymmetric Duffing equation under a parametric quasiperiodic perturbation.

Morozov A. D., Morozov K. E.
Degenerate Resonances and Synchronization in Nearly Hamiltonian Systems Under Quasiperiodic Perturbations
2022, vol. 27, no. 5, pp. 572585
Abstract
Quasiperiodic nonconservative perturbations of twodimensional nonlinear Hamiltonian
systems are considered. The definition of a degenerate resonance is introduced and
the topology of a degenerate resonance zone is studied. Particular attention is paid to the
synchronization process during the passage of an invariant torus through the resonance zone.
The existence of socalled synchronization intervals is proved and new phenomena which have
to do with synchronization are found. The study is based on the analysis of a pendulumtype
averaged system that determines the dynamics near the degenerate resonance phase curve of
the unperturbed system.

Karabanov A. A., Morozov A. D.
On Resonances in Hamiltonian Systems with Three Degrees of Freedom
2019, vol. 24, no. 6, pp. 628648
Abstract
We address the dynamics of nearintegrable Hamiltonian systems with 3 degrees
of freedom in extended vicinities of unperturbed resonant invariant Liouville tori. The main
attention is paid to the case where the unperturbed torus satisfies two independent resonance
conditions. In this case the average dynamics is 4dimensional, reduced to a generalised
motion under a conservative force on the 2torus and is generically nonintegrable. Methods of
differential topology are applied to full description of equilibrium states and phase foliations of
the average system. The results are illustrated by a simple model combining the nondegeneracy
and nonintegrability of the isoenergetically reduced system.

Morozov A. D.
On Bifurcations in Degenerate Resonance Zones
2014, vol. 19, no. 4, pp. 474482
Abstract
Bifurcations in degenerate resonance zones for Hamitonian systems with 3/2 degrees of freedom close to nonlinear integrable ones and for symplectic maps of a cylinder are discussed.

Howard J. E., Morozov A. D.
A Simple Reconnecting Map
2012, vol. 17, no. 5, pp. 417430
Abstract
Generalized standard maps of the cylinder for which the rotation number is a rational function (a combination of the Fermi and Chirikov rotation functions) are considered. These symplectic maps often have degenerate resonant zones, and we establish two types resonance bifurcations: "loops" and "vortex pairs". Both the border of chaos and the existence of the chaotic web are discussed. Finally the transition to global chaos for a generalized map is considered.

Morozov A. D., Mamedov E. A.
On a Double Cycle and Resonances
2012, vol. 17, no. 1, pp. 6371
Abstract
We consider systems with 3/2 degrees of freedom close to nonlinear autonomous Hamiltonian ones in the case where the perturbed autonomous systems have a double limit cycle. Then the initial nonautonomous systems have a special resonance zone. The structure of this zone is investigated.

Morozov A. D., Kondrashov R. E.
On Resonances in Systems of Two Weakly Coupled Oscillators
2009, vol. 14, no. 2, pp. 237247
Abstract
Assuming that two weakly coupled oscillators are essentially nonlinear we construct the most suitable form of a shortened 3dimensional system which describes behavior of solutions inside nondegenerate resonance zones. We analyze a model system of that kind and establish the existence of limit cycles of different types and also the existence of nonregular attractors which are explained by the existence of saddlefocus loops.

Morozov A. D.
On Degenerate Resonances and "Vortex Pairs"
2008, vol. 13, no. 1, pp. 2736
Abstract
Hamiltonian systems with 3/2 degrees of freedom close to nonlinear autonomous are studied. For unperturbed equations with a nonlinearity in the form of a polynomial of the fourth or fifth degree, their coefficients are specified for which the period on closed phase curves is not a monotone function of the energy and has extreme values of the maximal order. When the perturbation is periodic in time, this nonmonotonicity leads to the existence of degenerate resonances. The numerical study of the Poincaré map was carried out and bifurcations related to the formation of the vortex pairs within the resonance zones were found. For systems of a general form at arbitrarily small perturbations the absence of vortex pairs is proved. An explanation of the appearance of these structures for the Poincaré map is presented.

Karabanov A. A., Morozov A. D.
To the theory of coupled oscillations passing through the resonance
2006, vol. 11, no. 2, pp. 259268
Abstract
The problem of coupled oscillations is considered in the case when a stable equilibrium of a globally averaged system passes through a resonance curve. Questions of persistence of invariant tori and transition to a selfsynchronization are particularly discussed.

Morozov A. D.
On degenerate resonances in nearly Hamiltonian systems
2004, vol. 9, no. 3, pp. 337350
Abstract
The problem of topology of a neighborhood of degenerate resonance zone in timeperiodic twodimensional nearly Hamiltonian systems of differential equations is considered. The investigation is based on averaged systems analysis. The role of polyharmonical Hamiltonian and nonHamiltonian perturbations is discussed. In the Hamiltonian case, conditions of existence of degenerate resonances in nondegenerate and degenerate resonance zones are established. In the case of nonHamiltonian perturbations, a problem of synchronization on degenerate resonances is considered.

Morozov A. D.
On Hypercomplex Dynamics
2002, vol. 7, no. 3, pp. 281289
Abstract
Square mappings in a threedimensional hyperspace are considered. The properties of the closure of a set of repelling points are studied. In some sense, this closure is analogous to a Julia set. Computeraided visualizations of this set are given.

Morozov A. D., Boykova S. A.
On the investigation of degenerate resonances
1999, vol. 4, no. 1, pp. 7082
Abstract
For periodic in time systems, close to the twodimensional Hamiltonian ones, the problem of the topology of the neighbourhood of degenerate resonance levels is considered. The "truncated" system determining the topology of neighbourhood of degenerate level close to resonance level is conclusion. The behavior of the solutions of this system in dependence on the detuning is investigated and the bifurcations related to the transition from typical nonlinear resonance to degenerate resonance are determined (both in the case of impassable resonances and in the case of partly passable ones).

Morozov A. D., Dragunov T. N.
On Research of HenonHeilestype Systems
1997, vol. 2, no. 1, pp. 4354
Abstract
Threeparametrical family of systems with two degrees of freedom of a kind
$$\begin{array}{rcl}
\dfrac{d^2x_1}{dt^2}+x_1&=&2\varepsilon x_1x_2\\
\dfrac{d^2x_2}{dt^2}+x_2x_2^2&=&\varepsilon (x_1^2+\delta\dot{x}_2+\gamma x_2\dot{x}_2),
\end{array}\qquad\qquad(*)$$
where $\varepsilon>0$ is considered. Analytical research of trajectories behaviour of the system $(*)$ is carried out when $\varepsilon$ is small.
The given research is connected, first of all, to the analysis of resonant zones. Alongside with the initial system, another system $$\ddot{x}_2+x_2x_2^2=\varepsilon(A^2\sin^2t+\delta\dot{x}_2+\gamma x_2\dot{x}_2)\qquad\qquad(**)$$ that is "close" to the original, is considered. A good concurrence of results for Poincare mapping, induced by an equation $(**)$ when $\delta=\gamma=0$, and for the mapping that was constructed by Henon and Heiles, is established. In addition, for system $(*)$ a transition to nonregular dynamics is numerically analyzed at increase of parameter $\varepsilon$ and $\delta=\gamma=0$. It is established, that the transition to nonregular dynamics is connected, in particular, with the period doubling bifurcation (known as Feigenbaum's script), and $\varepsilon_\infty\approx0.95$. 