Albert Morozov

pr. Gagarina 23, Nizhny Novgorod, 603950 Russia
N.I. Lobachevsky State University of Nizhny Novgorod


Morozov K. E., Morozov A. D.
We study nonconservative quasi-periodic (with $m$ frequencies) perturbations of two-dimensional Hamiltonian systems with nonmonotonic rotation. It is assumed that the perturbation contains the so-called 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 quasi-periodic perturbation.
Keywords: nearly Hamiltonian system, degenerate resonance, quasi-periodic perturbation, parametric perturbation, averaging
Citation: Morozov K. E., Morozov A. D.,  Quasi-Periodic Parametric Perturbations of Two-Dimensional Hamiltonian Systems with Nonmonotonic Rotation, Regular and Chaotic Dynamics, 2024, vol. 29, no. 1, pp. 65-77
Morozov A. D., Morozov K. E.
Quasi-periodic nonconservative perturbations of two-dimensional 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 so-called synchronization intervals is proved and new phenomena which have to do with synchronization are found. The study is based on the analysis of a pendulum-type averaged system that determines the dynamics near the degenerate resonance phase curve of the unperturbed system.
Keywords: nearly Hamiltonian system, degenerate resonance, quasi-periodic perturbation, averaging, synchronization
Citation: Morozov A. D., Morozov K. E.,  Degenerate Resonances and Synchronization in Nearly Hamiltonian Systems Under Quasi-periodic Perturbations, Regular and Chaotic Dynamics, 2022, vol. 27, no. 5, pp. 572-585
Karabanov A. A., Morozov A. D.
We address the dynamics of near-integrable 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 4-dimensional, reduced to a generalised motion under a conservative force on the 2-torus and is generically non-integrable. 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 non-degeneracy and non-integrability of the isoenergetically reduced system.
Keywords: Hamiltonian systems, resonances, topological structures
Citation: Karabanov A. A., Morozov A. D.,  On Resonances in Hamiltonian Systems with Three Degrees of Freedom, Regular and Chaotic Dynamics, 2019, vol. 24, no. 6, pp. 628-648
Morozov A. D.
On Bifurcations in Degenerate Resonance Zones
2014, vol. 19, no. 4, pp.  474-482
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.
Keywords: resonances, degenerate resonances, bifurcations, Hamiltonian systems, averaged systems, separatrix, vortex pairs, symplectic maps
Citation: Morozov A. D.,  On Bifurcations in Degenerate Resonance Zones, Regular and Chaotic Dynamics, 2014, vol. 19, no. 4, pp. 474-482
Howard J. E., Morozov A. D.
A Simple Reconnecting Map
2012, vol. 17, no. 5, pp.  417-430
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.
Keywords: periodic points, degenerate resonances, vortex pairs, chaos
Citation: Howard J. E., Morozov A. D.,  A Simple Reconnecting Map, Regular and Chaotic Dynamics, 2012, vol. 17, no. 5, pp. 417-430
Morozov A. D., Mamedov E. A.
On a Double Cycle and Resonances
2012, vol. 17, no. 1, pp.  63-71
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 non-autonomous systems have a special resonance zone. The structure of this zone is investigated.
Keywords: cycle, resonances, average
Citation: Morozov A. D., Mamedov E. A.,  On a Double Cycle and Resonances, Regular and Chaotic Dynamics, 2012, vol. 17, no. 1, pp. 63-71
Morozov A. D., Kondrashov R. E.
On Resonances in Systems of Two Weakly Coupled Oscillators
2009, vol. 14, no. 2, pp.  237-247
Assuming that two weakly coupled oscillators are essentially nonlinear we construct the most suitable form of a shortened 3-dimensional system which describes behavior of solutions inside non-degenerate 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 saddle-focus loops.
Keywords: oscillators, resonances, cycles, equilibrium, attractors, average
Citation: Morozov A. D., Kondrashov R. E.,  On Resonances in Systems of Two Weakly Coupled Oscillators, Regular and Chaotic Dynamics, 2009, vol. 14, no. 2, pp. 237-247
Morozov A. D.
On Degenerate Resonances and "Vortex Pairs"
2008, vol. 13, no. 1, pp.  27-36
Hamiltonian systems with 3/2 degrees of freedom close to non-linear 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 non-monotonicity 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.
Keywords: Resonances, degenerate resonances, Hamiltonian systems, averaged systems, separatrix, vortex pairs, Poincaré map
Citation: Morozov A. D.,  On Degenerate Resonances and "Vortex Pairs", Regular and Chaotic Dynamics, 2008, vol. 13, no. 1, pp. 27-36
Karabanov A. A., Morozov A. D.
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 self-synchronization are particularly discussed.
Keywords: near-integrable systems, coupled oscillations, resonances, self-synchronization
Citation: Karabanov A. A., Morozov A. D.,  To the theory of coupled oscillations passing through the resonance , Regular and Chaotic Dynamics, 2006, vol. 11, no. 2, pp. 259-268
DOI: 10.1070/RD2006v011n02ABEH000349
Morozov A. D.
On degenerate resonances in nearly Hamiltonian systems
2004, vol. 9, no. 3, pp.  337-350
The problem of topology of a neighborhood of degenerate resonance zone in time-periodic two-dimensional nearly Hamiltonian systems of differential equations is considered. The investigation is based on averaged systems analysis. The role of polyharmonical Hamiltonian and non-Hamiltonian perturbations is discussed. In the Hamiltonian case, conditions of existence of degenerate resonances in non-degenerate and degenerate resonance zones are established. In the case of non-Hamiltonian perturbations, a problem of synchronization on degenerate resonances is considered.
Citation: Morozov A. D.,  On degenerate resonances in nearly Hamiltonian systems, Regular and Chaotic Dynamics, 2004, vol. 9, no. 3, pp. 337-350
Morozov A. D.
On Hypercomplex Dynamics
2002, vol. 7, no. 3, pp.  281-289
Square mappings in a three-dimensional 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. Computer-aided visualizations of this set are given.
Citation: Morozov A. D.,  On Hypercomplex Dynamics, Regular and Chaotic Dynamics, 2002, vol. 7, no. 3, pp. 281-289
Morozov A. D., Boykova S. A.
On the investigation of degenerate resonances
1999, vol. 4, no. 1, pp.  70-82
For periodic in time systems, close to the two-dimensional 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).
Citation: Morozov A. D., Boykova S. A.,  On the investigation of degenerate resonances, Regular and Chaotic Dynamics, 1999, vol. 4, no. 1, pp. 70-82
Morozov A. D., Dragunov T. N.
On Research of Henon-Heiles-type Systems
1997, vol. 2, no. 1, pp.  43-54
Three-parametrical family of systems with two degrees of freedom of a kind $$\begin{array}{rcl} \dfrac{d^2x_1}{dt^2}+x_1&=&-2\varepsilon x_1x_2\\ \dfrac{d^2x_2}{dt^2}+x_2-x_2^2&=&\varepsilon (-x_1^2+\delta\dot{x}_2+\gamma x_2\dot{x}_2), \end{array}\qquad\qquad(*)$$ 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 $$\ddot{x}_2+x_2-x_2^2=\varepsilon(-A^2\sin^2t+\delta\dot{x}_2+\gamma x_2\dot{x}_2)\qquad\qquad(**)$$ 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 $\varepsilon_\infty\approx0.95$.
Citation: Morozov A. D., Dragunov T. N.,  On Research of Henon-Heiles-type Systems, Regular and Chaotic Dynamics, 1997, vol. 2, no. 1, pp. 43-54

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