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2013
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# Albert Morozov

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

## Publications:

 Morozov A. D. On Bifurcations in Degenerate Resonance Zones 2014, vol. 19, no. 4, pp.  474-482 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. 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 DOI:10.1134/S1560354714040030
 Howard J. E., Morozov A. D. A Simple Reconnecting Map 2012, vol. 17, no. 5, pp.  417-430 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. 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 DOI:10.1134/S1560354712050048
 Morozov A. D., Mamedov E. A. On a Double Cycle and Resonances 2012, vol. 17, no. 1, pp.  63-71 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 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 DOI:10.1134/S1560354712010066
 Morozov A. D., Kondrashov R. E. On Resonances in Systems of Two Weakly Coupled Oscillators 2009, vol. 14, no. 2, pp.  237-247 Abstract 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 DOI:10.1134/S156035470902004X
 Morozov A. D. On Degenerate Resonances and "Vortex Pairs" 2008, vol. 13, no. 1, pp.  27-36 Abstract 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 DOI:10.1134/S1560354708010048
 Karabanov A. A., Morozov A. D. To the theory of coupled oscillations passing through the resonance 2006, vol. 11, no. 2, pp.  259-268 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 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 Abstract 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 DOI:10.1070/RD2004v009n03ABEH000283
 Morozov A. D. On Hypercomplex Dynamics 2002, vol. 7, no. 3, pp.  281-289 Abstract 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 DOI:10.1070/RD2002v007n03ABEH000210
 Morozov A. D., Boykova S. A. On the investigation of degenerate resonances 1999, vol. 4, no. 1, pp.  70-82 Abstract 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 DOI:10.1070/RD1999v004n01ABEH000099
 Morozov A. D., Dragunov T. N. On Research of Henon-Heiles-type Systems 1997, vol. 2, no. 1, pp.  43-54 Abstract 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 DOI:10.1070/RD1997v002n01ABEH000025