Vol. 13, No. 3

Vol. 13, No. 3, 2017

Doroshenko V. M.,  Kruglov V. P.,  Kuznetsov S. P.
A nonautonomous system with a uniformly hyperbolic attractor of Smale – Williams type in a Poincaré cross-section is proposed with generation implemented on the basis of the effect of oscillation death. The results of a numerical study of the system are presented: iteration diagrams for phases and portraits of the attractor in the stroboscopic Poincaré cross-section, power density spectra, Lyapunov exponents and their dependence on parameters, and the atlas of regimes. The hyperbolicity of the attractor is verified using the criterion of angles.
Keywords: uniformly hyperbolic attractor, Smale–Williams solenoid, Bernoulli map, oscillation death, Lyapunov exponents
Citation: Doroshenko V. M.,  Kruglov V. P.,  Kuznetsov S. P., Chaos generator with the Smale–Williams attractor based on oscillation death, Rus. J. Nonlin. Dyn., 2017, Vol. 13, No. 3, pp. 303-315
Shepelev I. A.,  Vadivasova T. E.
This paper is concerned with the spatiotemporal dynamics of the 2D lattice of cubic maps with nonlocal coupling. Different types of chimera structures have been found. Also, the underexplored regime of solitary states has been found. It is shown that the solitary states are typical of a large coupling radius. The possibility of detecting such a regime increases with the transition to global interaction, while chimera states disappear.
Keywords: oscillator ensemble, 2D lattice, nonlocal interaction, global coupling, spatial structure, chimera state, solitary state
Citation: Shepelev I. A.,  Vadivasova T. E., Solitary states in a 2D lattice of bistable elements with global and close to global interaction, Rus. J. Nonlin. Dyn., 2017, Vol. 13, No. 3, pp. 317-329
Isaeva O. B.,  Obychev M. A.,  Savin D. V.
An abstract discrete time dynamical system, given by an implicit function of the values of a variable at successive moments of time, is presented. The dynamics of this system is defined ambiguously both in reverse and forward time. An example of a system of such type is described in the works of Bullett, Osbaldestin and Percival [Physica D, 1986, vol. 19, pp. 290–300; Nonlinearity, 1988, vol. 1, pp. 27–50]; it demonstrates some features of the behavior of Hamiltonian systems. The map under study allows a smooth transition from the case of the explicitly defined evolution operator to an implicit one and, further, to the “conservative” limit, corresponding to the symmetric evolution operator satisfying the unitarity condition. Being created on the basis of the complex Mandelbrot map, it demonstrates the transformation of the phenomena of complex analytical dynamics to “conservative” phenomena and allows us to identify the relationship between them.
Keywords: Mandelbrot set, Julia set, conservative and quasi-conservative dynamics, multistability, implicit map
Citation: Isaeva O. B.,  Obychev M. A.,  Savin D. V., Dynamics of a discrete system with the operator of evolution given by an implicit function: from the Mandelbrot map to a unitary map, Rus. J. Nonlin. Dyn., 2017, Vol. 13, No. 3, pp. 331-348
Kornilov M. V.,  Sysoev I. V.
External low-frequency interference (including interference with a pronounced main frequency) is a common problem in measurements of complex signals, which can affect results of coupling estimation. Since it is impossible to completely remove the interference without affecting the signal itself, the question arises: what distorts the results of coupling estimation to a lesser extent: filtering the interference or ignoring it?
The Granger causality (GC) method is one of the most popular approaches to the detection of directional coupling from observed signals. GC uses predictive empirical models, mostly, linear and nonlinear autoregressive models (recurrence maps). Since the method is highly parametric, its success depends primarily on the parameters of the models and on the properties of the signals. Therefore, the method has to be adapted to the data. In physiology and climatology, most signals have a pronounced time scale, so one of the most important problems is that of adapting the Granger causality method to signals with a selected time scale.
The purpose of this paper is to formulate recommendations for using the Granger causality method for signals with a pronounced temporal scale in the presence of common low-frequency interference. In this paper, we restrict our attention to the case of testing for unilateral coupling and use the recommendations and criteria, developed earlier, for the effectiveness of the method. The sensitivity and specificity of the method are estimated based on surrogate time series. The testing is performed using reference systems of nonlinear dynamics and radiophysics.
It is shown that the loss of sensitivity and specificity of the method decrease nonlinearly with increasing amplitude of the total interference. This dependence varies for different parameters of the method. If the power of interference is several per cent of the signal power, the best results can be achieved by an appropriate choice of parameters of the method rather than by filtering the interference. At a higher noise power, filtering is preferable.
Keywords: time series, coupling analysis, Granger causality, low frequency interference
Citation: Kornilov M. V.,  Sysoev I. V., Estimating the efficiency of the Granger causality method for detecting unidirectional coupling in the presence of common low frequency interference, Rus. J. Nonlin. Dyn., 2017, Vol. 13, No. 3, pp. 349-362
Shlufman K. V.,  Neverova G. P.,  Frisman E. Y.
The paper studies dynamic modes of the Ricker model with the periodic Malthusian parameter. The equation parametric space is shown to have multistability areas in which different dynamic modes are possible depending on the initial conditions. In particular, the model trajectory can asymptotically tend either to a stable cycle or to a chaotic attractor. Oscillation synchronization of the 2-cycles and the Malthusian parameter of the model are studied. Fluctuations in population size and environmental factors can be either synchronous or asynchronous. The structural features of attraction basins in phase space are investigated for possible stable dynamic modes.
Keywords: recurrence equation, Ricker model, periodic Malthusian parameter, stability, bifurcation, dynamic modes, phase space, basins of attraction, multistability
Citation: Shlufman K. V.,  Neverova G. P.,  Frisman E. Y., Dynamic modes of the Ricker model with periodic Malthusian parameter, Rus. J. Nonlin. Dyn., 2017, Vol. 13, No. 3, pp. 363-380
Ishbulatov J. M.,  Karavaev A. S.,  Ponomarenko V. I.,  Kiselev A. R.,  Sergeev S. A.,  Seleznev Y. P.,  Bezrychko B. P.,  Prokhorov M. D.
We propose an original mathematical model for the human cardiovascular system. The model simulates the heart rate, autonomous control of heart, arterial pressure and cardiorespiratory interaction. Taking into account the self-excited autonomic control allowed us to reproduce the experimentally observed effects of phase synchronization between the control elements. The consistency of the proposed model is validated by quantitative and qualitative reproduction of spectral and statistical characteristics of real data from healthy subjects. Within physiological values of the parameters the model demonstrates chaotic dynamics and reproduces spontaneous interchange between the intervals of spontaneous and nonspontaneous behavior.
Keywords: mathematical model, synchronization, cardiovascular system, dynamic chaos, time delay system
Citation: Ishbulatov J. M.,  Karavaev A. S.,  Ponomarenko V. I.,  Kiselev A. R.,  Sergeev S. A.,  Seleznev Y. P.,  Bezrychko B. P.,  Prokhorov M. D., Phase synchronization of elements of autonomic control in mathematical model of cardiovascular system, Rus. J. Nonlin. Dyn., 2017, Vol. 13, No. 3, pp. 381-397
Zhuzhoma E. V.,  Medvedev V. S.,  Isaenkova N. V.
In this paper, using methods of Morse – Smale dynamical systems, we consider the topological structure of the magnetic field of regions of the photosphere for a point-charge model. For an arbitrary number of charges (regardless of their location), without assuming a potentiality of the field $\boldsymbol{\vec B}$ (and hence without applying specific formulas), we give estimates that connect the numbers of charges of a certain type with the numbers of null-points. For the boundary estimates, we describe the topological structure of the magnetic field. We present a bifurcation of the birth of a large number of separators.
Keywords: dynamical Morse–Smale system, null-points, separator
Citation: Zhuzhoma E. V.,  Medvedev V. S.,  Isaenkova N. V., On the topological structure of the magnetic field of regions of the photosphere, Rus. J. Nonlin. Dyn., 2017, Vol. 13, No. 3, pp. 399-412
Maslov D. A.,  Merkuryev I. V.
A wave solid-state gyroscope with a cylindrical resonator and electrostatic control sensors is considered. The gyroscope dynamics mathematical model describing nonlinear oscillations of the resonator under voltage on the electrodes is used. The reference voltage causes a cubic nonlinearity and the alternating voltage causes a quadratic nonlinearity of the control forces.
Various regimes of supplying voltage to gyro sensors are investigated. For the linearization of oscillations the form of voltages on the electrodes is presented. These voltages compensate for both nonlinear oscillations of the resonator caused by electrostatic sensors and those caused by other physical and geometric factors. It is shown that the control forces have a nonlinearity that is eliminated by the voltage applied to the electrode system according to a special law.
The proposed method can be used to eliminate nonlinear oscillations and to linearize power characteristics of sensors for controlling wave solid-state gyroscopes with hemispherical, cylindrical and ring resonators.
Keywords: wave solid-state gyroscope, cylindrical resonator, nonlinear oscillations
Citation: Maslov D. A.,  Merkuryev I. V., The linearization for wave solid-state gyroscope resonator oscillations and electrostatic control sensors forces, Rus. J. Nonlin. Dyn., 2017, Vol. 13, No. 3, pp. 413-421
Seregin S. V.
This paper investigates the dynamic characteristics of a simple estimation model of the imperfect resonator of a wave solid-state gyroscope — a circumferential element under planar deformation conditions with initial deviations from the perfect circular form. Particular examples are given to show that the splitting of the bending frequency spectrum of geometrically imperfect rings may arise in cases different from those presented in modern theory. A pattern was established in which the splitting of the bending frequency spectrum of an imperfect ring arises. The unbalancing of the frequency content occurs when the number of forming waves is equal to the number of waves of imperfection of the ring’s shape and when the number of forming waves is twice, three, four and more times larger than that of the waves of shape imperfection. If the number of waves of ring shape imperfection is even, then the splitting of the bending frequency spectrum occurs even in the case where the number of forming waves is half the number of waves of shape imperfection, and also in cases where the number of forming waves is one and a half times, twice, two and a half times, three and a half, and more times larger than that of waves of shape imperfection; in the former case the unbalancing of the frequency content can be very significant.
Keywords: ring, wave solid-state gyroscope, resonator, initial imperfections, vibrations, fission, frequency spectrum, resonance
Citation: Seregin S. V., The influence of shape imperfections on the vibrations of a ring resonator of a wave solid-state gyroscope, Rus. J. Nonlin. Dyn., 2017, Vol. 13, No. 3, pp. 423-431
Bizyaev I. A.,  Borisov A. V.,  Mamaev I. S.
This paper is concerned with the Hess case in the Euler–Poisson equations and with its generalization on the pencil of Poisson brackets. It is shown that in this case the problem reduces to investigating the vector field on a torus and that the graph showing the dependence of the rotation number on parameters has horizontal segments (limit cycles) only for integer values of the rotation number. In addition, an example of a Hamiltonian system is given which possesses an invariant submanifold (similar to the Hess case), but on which the dependence of the rotation number on parameters is a Cantor ladder.
Keywords: invariant submanifold, rotation number, Cantor ladder, limit cycles
Citation: Bizyaev I. A.,  Borisov A. V.,  Mamaev I. S., The Hess–Appelrot case and quantization of the rotation number, Rus. J. Nonlin. Dyn., 2017, Vol. 13, No. 3, pp. 433-452
Citation: Vladimir Vasil’evich Beletskii (02.05.1930 – 20.07.2017), Rus. J. Nonlin. Dyn., 2017, Vol. 13, No. 3, pp. 453-456

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