Topological monodromy as an obstruction to Hamiltonization of nonholonomic systems: Pro or contra?
Journal of Geometry and Physics, 2015, vol. 87, pp. 61-75
Abstract
pdf (754.64 Kb)
The phenomenon of a topological monodromy in integrable Hamiltonian and nonholonomic systems is discussed. An efficient method for computing and visualizing the monodromy is developed. The comparative analysis of the topological monodromy is given for the rolling ellipsoid of revolution problem in two cases, namely, on a smooth and on a rough plane. The first of these systems is Hamiltonian, the second is nonholonomic. We show that, from the viewpoint of monodromy, there is no difference between the two systems, and thus disprove the conjecture by Cushman and Duistermaat stating that the topological monodromy gives a topological obstruction for Hamiltonization of the rolling ellipsoid of revolution on a rough plane.
Bolsinov A. V., Kilin A. A., Kazakov A. O., Topological monodromy as an obstruction to Hamiltonization of nonholonomic systems: Pro or contra? , Journal of Geometry and Physics, 2015, vol. 87, pp. 61-75
Sequential Dynamics in the Motif of Excitatory Coupled Elements
Regular and Chaotic Dynamics, 2015, vol. 20, no. 6, pp. 701-715
Abstract
pdf (769.83 Kb)
In this article a new model of motif (small ensemble) of neuron-like elements is proposed. It is built with the use of the generalized Lotka?Volterra model with excitatory couplings. The main motivation for this work comes from the problems of neuroscience where excitatory couplings are proved to be the predominant type of interaction between neurons of the brain. In this paper it is shown that there are two modes depending on the type of coupling between the elements: the mode with a stable heteroclinic cycle and the mode with a stable limit cycle. Our second goal is to examine the chaotic dynamics of the generalized three-dimensional Lotka?Volterra model.
Korotkov A. G., Kazakov A. O., Osipov G. V., Sequential Dynamics in the Motif of Excitatory Coupled Elements, Regular and Chaotic Dynamics, 2015, vol. 20, no. 6, pp. 701-715
Dynamics of the Suslov Problem in a Gravitational Field: Reversal and Strange Attractors
Regular and Chaotic Dynamics, 2015, vol. 20, no. 5, pp. 605-626
Abstract
pdf (640.12 Kb)
In this paper, we present some results on chaotic dynamics in the Suslov problem which describe the motion of a heavy rigid body with a fixed point, subject to a nonholonomic constraint, which is expressed by the condition that the projection of angular velocity onto the body-fixed axis is equal to zero. Depending on the system parameters, we find cases of regular (in particular, integrable) behavior and detect various attracting sets (including strange attractors) that are typical of dissipative systems. We construct a chart of regimes with regions characterizing chaotic and regular regimes depending on the degree of conservativeness. We examine in detail the effect of reversal, which was observed previously in the motion of rattlebacks.
Bizyaev I. A., Borisov A. V., Kazakov A. O., Dynamics of the Suslov Problem in a Gravitational Field: Reversal and Strange Attractors, Regular and Chaotic Dynamics, 2015, vol. 20, no. 5, pp. 605-626
Nonlinear dynamics of the rattleback: a nonholonomic model
Physics-Uspekhi, 2014, vol. 184, no. 5, pp. 453-460
Abstract
pdf (750.09 Kb)
For a solid body of convex form moving on a rough horizontal plane that is known as a rattleback, numerical simulations are used to discuss and illustrate dynamical phenomena that are characteristic of the motion due to a nonholonomic nature of the mechanical system; the relevant feature is the nonconservation of the phase volume in the course of the dynamics. In such a system, a local compression of the phase volume can produce behavior features similar to those exhibited by dissipative systems, such as stable equilibrium points corresponding to stationary rotations; limit cycles (rotations with oscillations); and strange attractors. A chart of dynamical regimes is plotted in a plane whose axes are the total mechanical energy and the relative angle between the geometric and dynamic principal axes of the body. The transition to chaos through a sequence of Feigenbaum period doubling bifurcations is demonstrated. A number of strange attractors are considered, for which phase portraits, Lyapunov exponents, and Fourier spectra are presented.
Citation:
Borisov A. V., Kazakov A. O., Kuznetsov S. P., Nonlinear dynamics of the rattleback: a nonholonomic model , Physics-Uspekhi, 2014, vol. 184, no. 5, pp. 453-460
The Reversal and Chaotic Attractor in the Nonholonomic Model of Chaplygin?s Top
Regular and Chaotic Dynamics, 2014, vol. 19, no. 6, pp. 718-733
Abstract
pdf (1.29 Mb)
In this paper we consider the motion of a dynamically asymmetric unbalanced ball on a plane in a gravitational field. The point of contact of the ball with the plane is subject to a nonholonomic constraint which forbids slipping. The motion of the ball is governed by the nonholonomic reversible system of 6 differential equations. In the case of arbitrary displacement of the center of mass of the ball the system under consideration is a nonintegrable system without an invariant measure. Using qualitative and quantitative analysis we show that the unbalanced ball exhibits reversal (the phenomenon of reversal of the direction of rotation) for some parameter values. Moreover, by constructing charts of Lyaponov exponents we find a few types of strange attractors in the system, including the so-called figure-eight attractor which belongs to the genuine strange attractors of pseudohyperbolic type.
Keywords:
rolling without slipping, reversibility, involution, integrability, reversal, chart of Lyapunov exponents, strange attractor
Citation:
Borisov A. V., Kazakov A. O., Sataev I. R., The Reversal and Chaotic Attractor in the Nonholonomic Model of Chaplygin?s Top, Regular and Chaotic Dynamics, 2014, vol. 19, no. 6, pp. 718-733
Regular and Chaotic Attractors in the Nonholonomic Model of Chapygin's ball
Russian Journal of Nonlinear Dynamics, 2014, vol. 10, no. 3, pp. 361-380
Abstract
pdf (1.3 Mb)
We study both analytically and numerically the dynamics of an inhomogeneous ball on a rough horizontal plane under the infuence of gravity. A nonholonomic constraint of zero velocity at the point of contact of the ball with the plane is imposed. In the case of an arbitrary displacement of the center of mass of the ball, the system is nonintegrable without the property of phase volume conservation. We show that at certain parameter values the unbalanced ball exhibits the effect of reversal (the direction of the ball rotation reverses). Charts of dynamical regimes on the parameter plane are presented. The system under consideration exhibits diverse chaotic dynamics, in particular, the figure-eight chaotic attractor, which is a special type of pseudohyperbolic chaos.
Keywords:
Chaplygin?s top, rolling without slipping, reversibility, involution, integrability, reverse, chart of dynamical regimes, strange attractor
Citation:
Borisov A. V., Kazakov A. O., Sataev I. R., Regular and Chaotic Attractors in the Nonholonomic Model of Chapygin's ball, Russian Journal of Nonlinear Dynamics, 2014, vol. 10, no. 3, pp. 361-380
Bifurcations and chaos in the problem of the motion of two point vortices in an acoustic wave
Russian Journal of Nonlinear Dynamics, 2014, vol. 10, no. 3, pp. 329-343
Abstract
pdf (5.62 Mb)
This paper is concerned with the dynamics of two point vortices of the same intensity which are affected by an acoustic wave. Typical bifurcations of fixed points have been identified by constructing charts of dynamical regimes, and bifurcation diagrams have been plotted.
Keywords:
point vortices, nonintegrability, bifurcations, chart of dynamical regimes
Citation:
Vetchanin E. V., Kazakov A. O., Bifurcations and chaos in the problem of the motion of two point vortices in an acoustic wave, Russian Journal of Nonlinear Dynamics, 2014, vol. 10, no. 3, pp. 329-343
Richness of Chaotic Dynamics in Nonholonomic Models of a Celtic Stone
Regular and Chaotic Dynamics, 2013, vol. 18, no. 5, pp. 521-538
Abstract
pdf (3.11 Mb)
We study the regular and chaotic dynamics of two nonholonomic models of a Celtic stone. We show that in the first model (the so-called BM-model of a Celtic stone) the chaotic dynamics arises sharply, during a subcritical period doubling bifurcation of a stable limit cycle, and undergoes certain stages of development under the change of a parameter including the appearance of spiral (Shilnikov-like) strange attractors and mixed dynamics. For the second model, we prove (numerically) the existence of Lorenz-like attractors (we call them discrete Lorenz attractors) and trace both scenarios of development and break-down of these attractors.
Gonchenko A. S., Gonchenko S. V., Kazakov A. O., Richness of Chaotic Dynamics in Nonholonomic Models of a Celtic Stone, Regular and Chaotic Dynamics, 2013, vol. 18, no. 5, pp. 521-538
Strange Attractors and Mixed Dynamics in the Problem of an Unbalanced Rubber Ball Rolling on a Plane
Regular and Chaotic Dynamics, 2013, vol. 18, no. 5, pp. 508-520
Abstract
pdf (1.65 Mb)
We consider the dynamics of an unbalanced rubber ball rolling on a rough plane. The term rubber means that the vertical spinning of the ball is impossible. The roughness of the plane means that the ball moves without slipping. The motions of the ball are described by a nonholonomic system reversible with respect to several involutions whose number depends on the type of displacement of the center of mass. This system admits a set of first integrals, which helps to reduce its dimension. Thus, the use of an appropriate two-dimensional Poincaré map is enough to describe the dynamics of our system. We demonstrate for this system the existence of complex chaotic dynamics such as strange attractors and mixed dynamics. The type of chaotic behavior depends on the type of reversibility. In this paper we describe the development of a strange attractor and then its basic properties. After that we show the existence of another interesting type of chaos — the so-called mixed dynamics. In numerical experiments, a set of criteria by which the mixed dynamics may be distinguished from other types of dynamical chaos in two-dimensional maps is given.
Kazakov A. O., Strange Attractors and Mixed Dynamics in the Problem of an Unbalanced Rubber Ball Rolling on a Plane, Regular and Chaotic Dynamics, 2013, vol. 18, no. 5, pp. 508-520
Chaotic dynamics phenomena in the rubber rock-n-roller on a plane problem
Russian Journal of Nonlinear Dynamics, 2013, vol. 9, no. 2, pp. 309-325
Abstract
pdf (5.28 Mb)
Inšthis paper wešstudy ašproblem ofšrolling ofšthe dynamically asymmetric ball with displacement center ofšgravity onšašplane without slipping and vertical rotating. Itšisšshown that the dynamics ofšthe ball isšsignificantly affected byšthe type ofšreversibility. Depending onšthe type ofšthe reversibility wešfound two different types ofšdynamical chaos: strange attractors and mixed chaotic dynamics. Inšthis paper wešdescribe ašstrange attractor development, and then its basic properties. Ašset ofšcriteria byšwhich inšnumerical experiments mixed dynamics may bešdistinguished from other types ofšdynamical chaos are given.
Kazakov A. O., Chaotic dynamics phenomena in the rubber rock-n-roller on a plane problem, Russian Journal of Nonlinear Dynamics, 2013, vol. 9, no. 2, pp. 309-325
Integrability and stochastic behavior in some nonholonomic dynamics problems
Russian Journal of Nonlinear Dynamics, 2013, vol. 9, no. 2, pp. 257-265
Abstract
pdf (2.27 Mb)
Inšthis paper, wešinvestigate the dynamics ofšsystems describing the rolling without slipping and spinning (rubber rolling) ofšanšellipsoid onšašplane and ašsphere. Wešresearch these problems using Poincare maps, which investigation helps tošdiscover ašnew integrable case.
Keywords:
nonholonomic constraint, invariant measure, first integral, Poincare map, integrability and chaos
Citation:
Bizyaev I. A., Kazakov A. O., Integrability and stochastic behavior in some nonholonomic dynamics problems, Russian Journal of Nonlinear Dynamics, 2013, vol. 9, no. 2, pp. 257-265
Topological monodromy in nonholonomic systems
Russian Journal of Nonlinear Dynamics, 2013, vol. 9, no. 2, pp. 203-227
Abstract
pdf (890.26 Kb)
The phenomenon ofšaštopological monodromy inšintegrable Hamiltonian and nonholonomic systems isšdiscussed. Anšefficient method for computing and visualizing the monodromy isšdeveloped. The comparative analysis ofšthe topological monodromy isšgiven for the rolling ellipsoid ofšrevolution problem inštwo cases, namely, onšašsmooth and onšašrough plane. The first ofšthese systems isšHamiltonian, the second isšnonholonomic. Wešshow that, from the viewpoint ofšmonodromy, there isšnošdifference between the two systems, and thus disprove the conjecture byšCushman and Duistermaat stating that the topological monodromy gives aštopological obstruction for Hamiltonization ofšthe rolling ellipsoid ofšrevolution onšašrough plane.
Bolsinov A. V., Kilin A. A., Kazakov A. O., Topological monodromy in nonholonomic systems, Russian Journal of Nonlinear Dynamics, 2013, vol. 9, no. 2, pp. 203-227
On some new aspects of Celtic stone chaotic dynamics
Russian Journal of Nonlinear Dynamics, 2012, vol. 8, no. 3, pp. 507-518
Abstract
pdf (26.28 Mb)
Wešstudy chaotic dynamics ofšašnonholonomic model ofšceltic stone movement onšthe plane. Scenarious ofšappearance and development ofšchaos are investigated.
Gonchenko A. S., Gonchenko S. V., Kazakov A. O., On some new aspects of Celtic stone chaotic dynamics, Russian Journal of Nonlinear Dynamics, 2012, vol. 8, no. 3, pp. 507-518