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
pdf (2.91 Mb)
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
Landau–Hopf scenario in the ensemble of interacting oscillators
Russian Journal of Nonlinear Dynamics, 2012, vol. 8, no. 5, pp. 863-873
pdf (578.41 Kb)
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.
Phenomena of nonlinear dynamics of dissipative systems in nonholonomic mechanics of the rattleback
Russian Journal of Nonlinear Dynamics, 2012, vol. 8, no. 4, pp. 735-762
pdf (1.42 Mb)
We perform a numerical study of the motion of the rattleback, a rigid body with a convex surface on a rough horizontal plane in dependence on the parameters, applying the methods used previously for the treatment of dissipative dynamical systems, and adapted for the nonholonomic model. Charts of dynamical regimes are presented on the parameter plane of the total mechanical energy and the angle between the geometric and dynamic principal axes of the rigid body. Presence of characteristic structures in the parameter space, previously observed only for dissipative systems, is demonstrated. A method of calculating for the full spectrum of Lyapunov exponents is developed and implemented. It is shown that analysis of the Lyapunov exponents of chaotic regimes of the nonholonomic model reveals two classes, one of which is typical for relatively high energies, and the second for the relatively small energies. For the model reduced to a three-dimensional 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 the quasiconservative type, with close in magnitude positive and negative Lyapunov exponents, and the rest one about zero. The transition to chaos through a sequence of period-doubling bifurcations is illustrated, and the observed scaling corresponds to that intrinsic to the dissipative systems. A study of strange attractors is provided, in particularly, phase portraits are presented as well as the Lyapunov exponents, the Fourier spectra, the results of calculating the fractal dimensions.
Kuznetsov S. P., Jalnine A. Y., Sataev I. R., Sedova Y. V., Phenomena of nonlinear dynamics of dissipative systems in nonholonomic mechanics of the rattleback, Russian Journal of Nonlinear Dynamics, 2012, vol. 8, no. 4, pp. 735-762
Critical point of accumulation of fold-flip bifurcation points and critical quasi-attractor (the review and new results)
Russian Journal of Nonlinear Dynamics, 2008, vol. 4, no. 2, pp. 113-132
pdf (1.02 Mb)
In paper we suggest an example of system which dynamics is answered to conception of a «critical quasi-attractor». Besides the brief review of earlier obtained results the new results are presented, namely the illustrations of scaling for basins of attraction of elements of critical quasi-attractor, the renormalization group approach in the presence of additive uncorrelated noise, the calculation of universal constant responsible for the scaling regularities of the noise effect, the illustrations of transitions initialized by noise that are realized between coexisted attractors.
quasi-attractor, renormalization group method, type of criticality, bifurcation, scaling, noise
Kuznetsov A. P., Kuznetsov S. P., Sataev I. R., Sedova Y. V., Critical point of accumulation of fold-flip bifurcation points and critical quasi-attractor (the review and new results), Russian Journal of Nonlinear Dynamics, 2008, vol. 4, no. 2, pp. 113-132
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
pdf (1.08 Mb)
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