Vol. 13, No. 3
Vol. 13, No. 3, 2017
Doroshenko V. M., Kruglov V. P., Kuznetsov S. P.
Abstract
A nonautonomous system with a uniformly hyperbolic attractor of Smale – Williams type in
a Poincaré crosssection 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 crosssection, 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.

Shepelev I. A., Vadivasova T. E.
Abstract
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.

Isaeva O. B., Obychev M. A., Savin D. V.
Abstract
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.

Kornilov M. V., Sysoev I. V.
Abstract
External lowfrequency 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 lowfrequency 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. 
Shlufman K. V., Neverova G. P., Frisman E. Y.
Abstract
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 2cycles 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.

Ishbulatov J. M., Karavaev A. S., Ponomarenko V. I., Kiselev A. R., Sergeev S. A., Seleznev Y. P., Bezrychko B. P., Prokhorov M. D.
Abstract
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 selfexcited 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.

Zhuzhoma E. V., Medvedev V. S., Isaenkova N. V.
Abstract
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 pointcharge 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 nullpoints. 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.

Maslov D. A., Merkuryev I. V.
Abstract
A wave solidstate 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 solidstate gyroscopes with hemispherical, cylindrical and ring resonators. 
Seregin S. V.
Abstract
This paper investigates the dynamic characteristics of a simple estimation model of the imperfect
resonator of a wave solidstate 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.

Bizyaev I. A., Borisov A. V., Mamaev I. S.
Abstract
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.
