Spiral Chaos in the Nonholonomic Model of a Chaplygin Top
Regular and Chaotic Dynamics, 2016, vol. 21, no. 7-8, pp. 939-954
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This paper presents a numerical study of the chaotic dynamics of a dynamically asymmetric unbalanced ball (Chaplygin top) rolling on a plane. It is well known that the dynamics of such a system reduces to the investigation of a three-dimensional map, which in the general case has no smooth invariant measure. It is shown that homoclinic strange attractors of discrete spiral type (discrete Shilnikov type attractors) arise in this model for certain parameters. From the viewpoint of physical motions, the trace of the contact point of a Chaplygin top on a plane is studied for the case where the phase trajectory sweeps out a discrete spiral attractor. Using the analysis of the trajectory of this trace, a conclusion is drawn
about the influence of “strangeness” of the attractor on the motion pattern of the top.
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
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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.
rolling without slipping, reversibility, involution, integrability, reversal, chart of Lyapunov exponents, strange attractor
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
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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.
Chaplygin’s top, rolling without slipping, reversibility, involution, integrability, reverse, chart of dynamical regimes, strange attractor
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
On a bifurcation scenario of a birth of attractor of Smale–Williams type
Russian Journal of Nonlinear Dynamics, 2013, vol. 9, no. 2, pp. 267-294
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We describe one possible scenario of destruction or of a birth of the hyperbolic attractors considering the Smale—Williams solenoid as an example. The content of the transition observed under variation of the control parameter is the pairwise merge of the orbits belonging to the attractor and to the unstable invariant set on the border of the basin of attraction, in the course of the set of bifurcations of the saddle-node type. The transition is not a single event, but occupies a finite interval on the control parameter axis. In an extended space of the state variables and the control parameter this scenario can be regarded as a mutual transformation of the stable and unstable solenoids one to each other. Several model systems are discussed manifesting this scenario e.g. the specially designed iterative maps and the physically realizable system of coupled alternately activated non-autonomous van der Pol oscillators. Detailed studies of inherent features and of the related statistical and scaling properties of the scenario are provided.
Isaeva O. B., Kuznetsov S. P., Sataev I. R., Pikovsky A., On a bifurcation scenario of a birth of attractor of Smale–Williams type, Russian Journal of Nonlinear Dynamics, 2013, vol. 9, no. 2, pp. 267-294
Dynamical Phenomena Occurring due to Phase Volume Compression in Nonholonomic Model of the Rattleback
Regular and Chaotic Dynamics, 2012, vol. 17, no. 6, pp. 512-532
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We study numerically the dynamics of the rattleback, a rigid body with a convex surface on a rough horizontal plane, in dependence on the parameters, applying methods used earlier for treatment of dissipative dynamical systems, and adapted here for the nonholonomic model. Charts of dynamical regimes on the parameter plane of the total mechanical energy and the angle between the geometric and dynamic principal axes of the rigid body are presented. Characteristic structures in the parameter space, previously observed only for dissipative systems, are revealed. A method for calculating the full spectrum of Lyapunov exponents is developed and implemented. Analysis of the Lyapunov exponents of the nonholonomic model reveals two classes of chaotic regimes. For the model reduced to a 3D map, the first one corresponds to a strange attractor with one positive and two negative Lyapunov exponents, and the second to the chaotic dynamics of quasi-conservative type, when positive and negative Lyapunov exponents are close in magnitude, and the remaining exponent is close to zero. The transition to chaos through a sequence of period-doubling bifurcations relating to the Feigenbaum universality class is illustrated. Several examples of strange attractors are considered in detail. In particular, phase portraits as well as the Lyapunov exponents, the Fourier spectra, and fractal dimensions are presented.
Borisov A. V., Jalnine A. Y., Kuznetsov S. P., Sataev I. R., Sedova Y. V., Dynamical Phenomena Occurring due to Phase Volume Compression in Nonholonomic Model of the Rattleback, Regular and Chaotic Dynamics, 2012, vol. 17, no. 6, pp. 512-532
Landau–Hopf scenario in the ensemble of interacting oscillators
Russian Journal of Nonlinear Dynamics, 2012, vol. 8, no. 5, pp. 863-873
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The conditions are discussed for which the ensemble of interacting oscillators may demonstrate Landau–Hopf scenario of successive birth of multi-frequency regimes. A model is proposed in the form of a network of five globally coupled oscillators, characterized by varying degree of excitement of individual oscillators. Illustrations are given for the birth of the tori of increasing dimension by successive quasi-periodic Hopf bifurcation.
Codimension and typicity in a context of description of transition to chaos via period-doubling in dissipative dynamical systems
Regular and Chaotic Dynamics, 1997, vol. 2, no. 3-4, pp. 90-105
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While considering multiparameter families of nonlinear systems, types of behavior at the onset of chaos may appear which are distinct from Feigenbaum's universality. We present a review of such situations which can be met in families of one-dimensional maps and discuss a possibility of their realization and observation in nonlinear dissipative systems of more general form.
Kuznetsov A. P., Kuznetsov S. P., Sataev I. R., Codimension and typicity in a context of description of transition to chaos via period-doubling in dissipative dynamical systems, Regular and Chaotic Dynamics, 1997, vol. 2, no. 3-4, pp. 90-105