Vol. 8, No. 3

Vol. 8, No. 3, 2012

Anishchenko V. S.,  Semenova N. I.,  Astakhov S. V.,  Boev Y. I.
The concept of a local fractal dimension has been introduced in the framework of the average Poincaré recurrence time numerical analysis in an $\varepsilon$-vicinity of a certain point. Lozi and Hénon maps have been considered. It has been shown that in case of Lozi map the local dimension weakly depends on the point on the attractor and its value is close to the fractal dimension of the attractor. In case of a quasi attractor observed in both Hénon and Feugenbaum systems the local dimension significantly depends on both the diameter and the location of the $\varepsilon$-vicinity. The reason of this strong dependency is high non-homogenity of a quasi-attractor which is typical for non-hyperbolic chaotic attractors.
Keywords: Poincaré recurrence, attractor dimension
Citation: Anishchenko V. S.,  Semenova N. I.,  Astakhov S. V.,  Boev Y. I., Poincaré recurrences time and local dimension of chaotic attractors, Rus. J. Nonlin. Dyn., 2012, Vol. 8, No. 3, pp. 449-460
Kuznetsov A. P.,  Kuznetsov S. P.,  Pozdnyakov M. V.,  Sedova Y. V.
We suggest a simple two-dimensional map, parameters of which are the trace and Jacobian of the perturbation matrix of the fixed point. On the parameters plane it demonstrates the main universal bifurcation scenarios: the threshold to chaos via period-doublings, the situation of quasiperiodic oscillations and Arnold tongues. We demonstrate the possibility of implementation of such map in radiophysical device.
Keywords: maps, bifurcations, phenomena of quasiperiodicity
Citation: Kuznetsov A. P.,  Kuznetsov S. P.,  Pozdnyakov M. V.,  Sedova Y. V., Universal two-dimensional map and its radiophysical realization, Rus. J. Nonlin. Dyn., 2012, Vol. 8, No. 3, pp. 461-471
Kuznetsov A. P.,  Pozdnyakov M. V.,  Sedova Y. V.
We examine the dynamics of the coupled system consisting of subsystems, demonstrating the Neimark–Sacker bifurcation. The study of coupled maps on the plane of the parameters responsible for such bifurcation in the individual subsystems is realized. On the plane of parameters characterizing the rotation numbers of the individual subsystems we reveal the complex structures consisting of the quasi-periodic modes of different dimensions and the exact periodic resonances of different orders.
Keywords: maps, bifurcations, phenomena of quasiperiodicity
Citation: Kuznetsov A. P.,  Pozdnyakov M. V.,  Sedova Y. V., Coupled universal maps demonstrating Neimark–Saker bifurcation, Rus. J. Nonlin. Dyn., 2012, Vol. 8, No. 3, pp. 473-482
Prokopenko V. G.
This paper is concerned with a method for forming compound (composite) chaotic multiattractors using the Lorentz equations as an example. These multiattractors are a union of several local attractors which are copies of some initial chaotic attractor.
Keywords: nonlinear dynamic system, autostochastic system, chaotic attractor, Lorentz attractor, compound multiattractor, metastable attractor, multisegment nonlinearity, intermittency, chaotic switchings, reduplication operator
Citation: Prokopenko V. G., Reduplication of chaotic attractors and construction of compound multiattractors, Rus. J. Nonlin. Dyn., 2012, Vol. 8, No. 3, pp. 483-496
Slepnev A. V.,  Vadivasova T. E.
The model of an active medium with periodical boundary conditions is studied. The elementary cell is chosen to be FitzHugh–Nagumo oscillator. According to the values of parameters the elementary cell is able to be either in self-sustained regime or in excitable one. In both cases there are sustained oscillations in each elementary cell of the medium, but the causes of its initiation are different. In case of the former each cell in itself is auto-oscillator, in case of the latter the oscillations appear because of feedback which is provided by the periodical boundary conditions. In both cases the phenomenon of multistability is observed. The comparative analysis of the regimes mentioned above is carried out. There are shown that the dependencies of oscillations characteristics from the system parameters in either cases significantly differ from one another. The bifurcational type of the transition from one cell regime to another is ascertained for some modes. The influence of spatial-uncorrelated noise on the active medium behavior is considered. The average period of oscillations versus noise intensity relation is obtained.
Keywords: active medium, FitzHugh–Nagumo system, spatial structures, multistability, noise influence
Citation: Slepnev A. V.,  Vadivasova T. E., Two kinds of auto-oscillations in active medium with periodical border conditions, Rus. J. Nonlin. Dyn., 2012, Vol. 8, No. 3, pp. 497-505
Gonchenko A. S.,  Gonchenko S. V.,  Kazakov A. O.
We study chaotic dynamics of a nonholonomic model of celtic stone movement on the plane. Scenarious of appearance and development of chaos are investigated.
Keywords: nonholonomic model, strange attractor, symmetry, bifurcation, mixed dynamics
Citation: Gonchenko A. S.,  Gonchenko S. V.,  Kazakov A. O., On some new aspects of Celtic stone chaotic dynamics, Rus. J. Nonlin. Dyn., 2012, Vol. 8, No. 3, pp. 507-518
Davletshin M. N.
In this article we deal with equations of a point motion on a surface in the Hamiltonian form in redundant coordinates. We also give explicit formulae of the Poisson’s bracket.
Keywords: hamiltonian systems, Poisson’s bracket, equations in surplus coordinates
Citation: Davletshin M. N., A Poisson’s bracket of a point motion on a surface, Rus. J. Nonlin. Dyn., 2012, Vol. 8, No. 3, pp. 519-522
Ol'shanskii V. Y.
For Poincaré–Zhukovsky’s equations with non-diagonal matrices in the Hamiltonian, we obtain conditions for existence of the quadratic integral $({\bf YS},{\bf K}) = \rm{const}$ and the explisit form of it. It is shown that if the integral exists, then the equations reduce to the Schottky’s case.
Keywords: Poincare–Zhukovsky’s equations, quadratic integral, non-diagonal matrices, Schottky’s case
Citation: Ol'shanskii V. Y., On quadratic integral Poincare–Zhukovsky’s equations, Rus. J. Nonlin. Dyn., 2012, Vol. 8, No. 3, pp. 523-540
Tsiganov A. V.
We prove the trajectory equivalence of the Chaplygin sphere problem, the Veselova system on $e^*(3)$ and a Hamiltonian system on two-dimensional sphere with the non-standard metric.
Keywords: nonholonomic systems, Poisson brackets
Citation: Tsiganov A. V., On the nonholonomic Veselova and Chaplygin systems, Rus. J. Nonlin. Dyn., 2012, Vol. 8, No. 3, pp. 541-547
Kozlov V. V.
We develop a new method for solving Hamilton’s canonical differential equations. The method is based on the search of invariant vortex manifolds of special type. In the case of Lagrangian (potential) manifolds, we arrive at the classical Hamilton–Jacobi method.
Keywords: generalized Lamb’s equations, vortex manifolds, Clebsch potentials, Lagrange brackets
Citation: Kozlov V. V., An extended Hamilton–Jacobi method, Rus. J. Nonlin. Dyn., 2012, Vol. 8, No. 3, pp. 549-568
Bizyaev I. A.,  Tsiganov A. V.
We discuss an embedding of the vector field associated with the nonholonomic Routh sphere in subgroup of the commuting Hamiltonian vector fields associated with this system. We prove that the corresponding Poisson brackets are reduced to canonical ones in the region without of homoclinic trajectories.
Keywords: nonholonomic mechanics, Routh sphere, Poisson brackets
Citation: Bizyaev I. A.,  Tsiganov A. V., On the Routh sphere, Rus. J. Nonlin. Dyn., 2012, Vol. 8, No. 3, pp. 569-583
Treschev D. V.,  Erdakova N. N.,  Ivanova T. B.
The problem of a uniform straight cylinder (disc) sliding on a horizontal plane under the action of dry friction forces is considered. The contact patch between the cylinder and the plane coincides with the base of the cylinder. We consider axisymmetric discs, i.e. we assume that the base of the cylinder is symmetric with respect to the axis lying in the plane of the base. The focus is on the qualitative properties of the dynamics of discs whose circular base, triangular base and three points are in contact with a rough plane.
Keywords: Amontons–Coulomb law, dry friction, disc, final dynamics, stability
Citation: Treschev D. V.,  Erdakova N. N.,  Ivanova T. B., On the final motion of cylindrical solids on a rough plane, Rus. J. Nonlin. Dyn., 2012, Vol. 8, No. 3, pp. 585-603
Bolsinov A. V.,  Borisov A. V.,  Mamaev I. S.
In the paper we consider a system of a ball that rolls without slipping on a plane. The ball is assumed to be inhomogeneous and its center of mass does not necessarily coincide with its geometric center. We have proved that the governing equations can be recast into a system of six ODEs that admits four integrals of motion. Thus, the phase space of the system is foliated by invariant 2-tori; moreover, this foliation is equivalent to the Liouville foliation encountered in the case of Euler of the rigid body dynamics. However, the system cannot be solved in terms of quadratures because there is no invariant measure which we proved by finding limit cycles.
Keywords: non-holonomic constraint, Liouville foliation, invariant torus, invariant measure, integrability
Citation: Bolsinov A. V.,  Borisov A. V.,  Mamaev I. S., Rolling without spinning of a ball on a plane: absence of an invariant measure in a system with a complete set of first integrals, Rus. J. Nonlin. Dyn., 2012, Vol. 8, No. 3, pp. 605-616
Sokolov S. V.,  Ramodanov S. M.
We consider a system which consists of a heavy circular cylinder and a point vortex in an unbounded volume of ideal liquid. The liquid is assumed to be irrotational and at rest at infinity. The circulation about the cylinder is different from zero. The governing equations are Hamiltonian and admit an evident integral of motion — the horizontal component of the momentum. Using the integral we reduce the order and thereby obtain a system with two degrees of freedom. Most remarkable types of partial solutions of the system are presented and stability of equilibrium solutions is investigated.
Keywords: point vortices, Hamiltonian systems, reduction, stability of equilibrium solutions
Citation: Sokolov S. V.,  Ramodanov S. M., Falling motion of a circular cylinder interacting dynamically with a point vortex, Rus. J. Nonlin. Dyn., 2012, Vol. 8, No. 3, pp. 617-628
Rybkin K. A.
The results of experiments on the free gravitational drift of light cylindrical bodies in the air. Based on the reconstruction of the dynamics of the time series with the construction of phase portraits in Takkensaspace, computing the spectrum of Lyapunov and Kolmogorov–Sinai entropy defined type of dynamics taking place processes and its basic parameters. The connection between the observed in experiments flicker noise hereditary process.
Keywords: drift bodies, flicker noise, the spectrum of Lyapunov
Citation: Rybkin K. A., Flicker noise in free fall cylinders in air, Rus. J. Nonlin. Dyn., 2012, Vol. 8, No. 3, pp. 629-639
Citation: On the 100th anniversary of the death of Henri Poincaré, Rus. J. Nonlin. Dyn., 2012, Vol. 8, No. 3, pp. 641-643
Chenciner A.
Citation: Chenciner A., Eloge de Poincaré, Rus. J. Nonlin. Dyn., 2012, Vol. 8, No. 3, pp. 645-651
Darboux G.
Citation: Darboux G., Henri Poincaré, Rus. J. Nonlin. Dyn., 2012, Vol. 8, No. 3, pp. 653-672
Citation: New books, Rus. J. Nonlin. Dyn., 2012, Vol. 8, No. 3, pp. 673-675

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