Alexey Jalnine
Zelenaya 38, Saratov, 410019, Russia
Saratov Branch of Kotelnikovâ€™s Institute of RadioEngineering and Electronics of RAS
Publications:
Jalnine A. Y., Kuznetsov S. P.
Autonomous Strange Nonchaotic Oscillations in a System of Mechanical Rotators
2017, vol. 22, no. 3, pp. 210225
Abstract
We investigate strange nonchaotic selfoscillations in a dissipative system consisting of three mechanical rotators driven by a constant torque applied to one of them. The external driving is nonoscillatory; the incommensurable frequency ratio in vibrationalrotational
dynamics arises due to an irrational ratio of diameters of the rotating elements involved. It is shown that, when losing stable equilibrium, the system can demonstrate two or threefrequency quasiperiodic, chaotic and strange nonchaotic selfoscillations. The conclusions of the work are confirmed by numerical calculations of Lyapunov exponents, fractal dimensions, spectral analysis, and by special methods of detection of a strange nonchaotic attractor (SNA): phase sensitivity and analysis using rational approximation for the frequency ratio. In particular, SNA possesses a zero value of the largest Lyapunov exponent (and negative values of the other exponents), a capacitive dimension close to 2 and a singular continuous power spectrum. In general, the results of this work shed a new light on the occurrence of strange nonchaotic dynamics.

Jalnine A. Y.
Generalized Synchronization of Identical Chaotic Systems on the Route from an Independent Dynamics to the Complete Synchrony
2013, vol. 18, no. 3, pp. 214225
Abstract
The transition from asynchronous hyperchaos to complete synchrony in coupled identical chaotic systems may either occur directly or be mediated by a preliminary stage of generalized synchronization. In the present paper we investigate the underlying mechanisms of realization of the both scenarios. It is shown that a generalized synchronization arises when the manifold of identically synchronous states $M$ is transversally unstable, while the local transversal contraction of phase volume first appears in the areas of phase space separated from $M$ and being visited by the chaotic trajectories. On the other hand, a direct transition from an asynchronous hyperchaos to the complete synchronization occurs, under variation of the controlling parameter, if the transversal stability appears first on the manifold $M$, and only then it extends upon the neighboring phase volume. The realization of one or another scenario depends upon the choice of the coupling function. This result is valid for both unidirectionally and mutually coupled systems, that is confirmed by theoretical analysis of the discrete models and numerical simulations of the physically realistic flow systems.

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
2012, vol. 17, no. 6, pp. 512532
Abstract
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 quasiconservative 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 perioddoubling 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.

Jalnine A. Y., Kuznetsov S. P., Osbaldestin A. H.
Dynamics of small perturbations of orbits on a torus in a quasiperiodically forced 2D dissipative map
2006, vol. 11, no. 1, pp. 1930
Abstract
We consider the dynamics of small perturbations of stable twofrequency quasiperiodic orbits on an attracting torus in the quasiperiodically forced Hénon map. Such dynamics consists in an exponential decay of the radial component and in a complex behaviour of the angle component. This behaviour may be two or threefrequency quasiperiodicity, or it may be irregular. In the latter case a graphic image of the dynamics of the perturbation angle is a fractal object, namely a strange nonchaotic attractor, which appears in auxiliary map for the angle component. Therefore, we claim that stable trajectories may approach the attracting torus either in a regular or in an irregular way. We show that the transition from quasiperiodic dynamics to chaos in the model system is preceded by the appearance of an irregular behaviour in the approach of the perturbed quasiperiodic trajectories to the smooth attracting torus. We also demonstrate a link between the evolution operator of the perturbation angle and a quasiperiodically forced circle mapping of a special form and with a Harper equation with quasiperiodic potential.
