Sergey Kuznetsov
S.P.Kuznetsov was born in 1951 in Moscow. In 1968 was graduated with gold medal from Saratov high school No 13, specialized in physics and mathematics (now the PhysicalTechnical Lyceum No 1), and enter the Saratov State University (Physical Department, Chair of Electronics). The diploma work was performed under guidance of Prof. D.I.Trubetskov.
Been graduated from the University in 1973, S.P.Kuznetsov started to work as an engineer in the Institute of Mechanics and Physics of SSU. In 19741977 he is a postgraduate student of Saratov University. In 1977 he received the degree of Candidate of Sciences (analog of PhD) from Saratov University. The title of the candidate thesis: "Theoretical methods for analysis of nonstationary phenomena in certain extended selfoscillating systems of interacting electron beam and electromagnetic wave", speciality Radiophysics. From 1977 till 1988 S.P.Kuznetsov is a Senior Researcher of the Institute of Mechanics and Physics of SSU. In 1984 he accepted the academic status of Senior Researcher, and in 1988 received degree of Doctor of Sciences from Saratov University. The title of the thesis: "Nonstationary nonlinear processes and stochastic oscillations in spatially extended systems of radiophysics and electronics". From 1988 S.P.Kuznetsov is a Head Researcher of Saratov Branch ofKotel'nikov's Institute of RadioEngineering and Electronics of RAS. In parallel, in 1992  1995 he is a Professor of Chair of Radiophysics and Nonlinear Dynamics of Saratov University, from 1996  a Professor of College of Applied Sciences, now Department of Nonlinear Processes of Saratov University. From 2001 S.P.Kuznetsov is a head of Laboratory of Theoretical Nonlinear Dynamics of SB IRE RAS, and from 2012 he is a head researcher of Laboratory of Nonlinear Analysis and Design of New Types of Vehicles (the Udmurt State University). S.P.Kuznetsov is an author of more than 200 published articles in Russian and International research journals, he has 3 inventor’s certificates. 10 candidate works (equivalent of PhD) have been performed under supervision of S.P.Kuznetsov.
Awards:
19941996: S.P.Kuznetsov was a Laureate of State Stipendium for Distinguished Scientists of Russian Federation
1988: Soros Associated Professor
2000, 2001: Soros Professor
Publications:
Kuznetsov S. P.
Some Lattice Models with Hyperbolic Chaotic Attractors
2020, Vol. 16, no. 1, pp. 1321
Abstract
Examples of onedimensional lattice systems are considered, in which patterns of different
spatial scales arise alternately, so that the spatial phase over a full cycle undergoes transformation
according to an expanding circle map that implies the occurrence of Smale–Williams attractors
in the multidimensional state space. These models can serve as a basis for design electronic
generators of robust chaos within a paradigm of coupled cellular networks. One of the examples
is a mechanical pendulum system interesting and demonstrative for research and educational
experimental studies.

Kuznetsov S. P., Kruglov V. P., Sedova Y. V.
Mechanical Systems with Hyperbolic Chaotic Attractors Based on Froude Pendulums
2020, Vol. 16, no. 1, pp. 5158
Abstract
We discuss two mechanical systems with hyperbolic chaotic attractors of Smale – Williams
type. Both models are based on Froude pendulums. The first system is composed of two
coupled Froude pendulums with alternating periodic braking. The second system is Froude
pendulum with timedelayed feedback and periodic braking. We demonstrate by means of
numerical simulations that the proposed models have chaotic attractors of Smale – Williams
type. We specify regions of parameter values at which the dynamics corresponds to the Smale –
Williams solenoid. We check numerically the hyperbolicity of the attractors.

Kuznetsov S. P.
Complex Dynamics in Generalizations of the Chaplygin Sleigh
2019, Vol. 15, no. 4, pp. 551559
Abstract
The article considers the Chaplygin sleigh on a plane in a potential well, assuming that
an external potential force is supplied at the mass center. Two particular cases are studied in
some detail, namely, a onedimensional potential valley and a potential with rotational symmetry;
in both cases the models reduce to fourdimensional differential equations conserving
mechanical energy. Assuming the potential functions to be quadratic, various behaviors are observed
numerically depending on the energy, from those characteristic to conservative dynamics
(regularity islands and chaotic sea) to strange attractors. This is another example of a nonholonomic
system manifesting these phenomena (similar to those for Celtic stone or Chaplygin
top), which reflects a fundamental nature of these systems occupying an intermediate position
between conservative and dissipative dynamics.

Kuznetsov S. P.
Generation of Robust Hyperbolic Chaos in CNN
2019, Vol. 15, no. 2, pp. 109124
Abstract
It is shown that on the basis of a cellular neural network (CNN) composed, e.g., of six cells, it is possible to design a chaos generator with an attractor being a kind of Smale – Williams solenoid, which provides chaotic dynamics that is rough (structurally stable), as follows from
respective fundamental mathematical theory. In the context of the technical device, it implies insensitivity to small variations of parameters, manufacturing imperfections, interferences, etc. Results of numerical simulations and circuit simulation in the Multisim environment are presented.
The proposed circuit is the first example of an electronic system where the role of the angular coordinate for the Smale – Williams attractor is played by the spatial phase of the sequence of patterns. It contributes to the collection of feasible systems with hyperbolic attractors and thus promotes filling with real content and promises practical application for the hyperbolic theory, which is an important and deep sector of the modern mathematical theory of dynamical systems.

Doroshenko V. M., Kruglov V. P., Kuznetsov S. P.
Smale – Williams Solenoids in a System of Coupled Bonhoeffer – van der Pol Oscillators
2018, Vol. 14, no. 4, pp. 435451
Abstract
The principle of constructing a new class of systems demonstrating hyperbolic chaotic attractors
is proposed. It is based on using subsystems, the transfer of oscillatory excitation
between which is provided resonantly due to the difference in the frequencies of small and large
(relaxation) oscillations by an integer number of times, accompanied by phase transformation
according to an expanding circle map. As an example, we consider a system where a Smale – Williams attractor is realized, which is based on two coupled Bonhoeffer – van der Pol oscillators.
Due to the applied modulation of parameter controlling the Andronov – Hopf bifurcation, the
oscillators manifest activity and suppression turn by turn. With appropriate selection of the
modulation form, relaxation oscillations occur at the end of each activity stage, the fundamental
frequency of which is by an integer factor $M = 2, 3, 4, \ldots$ smaller than that of small oscillations.
When the partner oscillator enters the activity stage, the oscillations start being stimulated by
the $M$th harmonic of the relaxation oscillations, so that the transformation of the oscillation
phase during the modulation period corresponds to the $M$fold expanding circle map. In the state
space of the Poincaré map this corresponds to an attractor of Smale – Williams type, constructed
with $M$fold increase in the number of turns of the winding at each step of the mapping. The results
of numerical studies confirming the occurrence of the hyperbolic attractors in certain parameter
domains are presented, including the waveforms of the oscillations, portraits of attractors, diagrams illustrating the phase transformation according to the expanding circle map, Lyapunov
exponents, and charts of dynamic regimes in parameter planes. The hyperbolic nature of the
attractors is verified by numerical calculations that confirm the absence of tangencies of stable
and unstable manifolds for trajectories on the attractor (“criterion of angles”). An electronic
circuit is proposed that implements this principle of obtaining the hyperbolic chaos and its
functioning is demonstrated using the software package Multisim.

Doroshenko V. M., Kruglov V. P., Kuznetsov S. P.
Chaos generator with the Smale–Williams attractor based on oscillation death
2017, Vol. 13, No. 3, pp. 303315
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.

Jalnine A. Y., Kuznetsov S. P.
Autonomous strange nonchaotic oscillations in a system of mechanical rotators
2017, Vol. 13, No. 2, pp. 257275
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.

Kuznetsov A. P., Kuznetsov S. P., Sedova Y. V.
Pendulum system with an infinite number of equilibrium states and quasiperiodic dynamics
2016, Vol. 12, No. 2, pp. 223234
Abstract
Examples of mechanical systems are discussed, where quasiperiodic motions may occur, caused by an irrational ratio of the radii of rotating elements that constitute the system. For the pendulum system with frictional transmission of rotation between the elements, in the conservative and dissipative cases we note the coexistence of an infinite number of stable fixed points, and in the case of the selfoscillating system the presence of many attractors in the form of limit cycles and of quasiperiodic rotational modes is observed. In the case of quasiperiodic dynamics the frequencies of spectral components depend on the parameters, but the ratio of basic incommensurate frequencies remains constant and is determined by the irrational number characterizing the relative size of the elements.

Kuznetsov S. P.
Hyperbolic chaos in selfoscillating systems based on mechanical triple linkage: Testing absence of tangencies of stable and unstable manifolds for phase trajectories
2016, Vol. 12, No. 1, pp. 121143
Abstract
Dynamical equations are formulated and a numerical study is provided for selfoscillatory model systems based on the triple linkage hinge mechanism of Thurston–Weeks–Hunt–MacKay. We consider systems with a holonomic mechanical constraint of three rotators as well as systems, where three rotators interact by potential forces. We present and discuss some quantitative characteristics of the chaotic regimes (Lyapunov exponents, power spectrum). Chaotic dynamics of the models we consider are associated with hyperbolic attractors, at least, at relatively small supercriticality of the selfoscillating modes; that follows from numerical analysis of the distribution for angles of intersection of stable and unstable manifolds of phase trajectories on the attractors. In systems based on rotators with interacting potential the hyperbolicity is violated starting from a certain level of excitation.

Kuznetsov S. P.
Motion of a falling card in a fluid: Finitedimensional models, complex phenomena, and nonlinear dynamics
2015, Vol. 11, No. 1, pp. 349
Abstract
Results are reviewed relating to the planar problem for the falling card in a resisting medium based on models represented by ordinary differential equations for a small number of variables. We introduce a unified model, which gives an opportunity to conduct a comparative analysis of dynamic behaviors of models of Kozlov, Tanabe – Kaneko, Belmonte – Eisenberg – Moses and Andersen – Pesavento – Wang using common dimensionless variables and parameters. It is shown that the overall structure of the parameter spaces for the different models shows certain similarities caused obviously by the same inherent symmetry and by universal nature of the involved phenomena of nonlinear dynamics (fixed points, limit cycles, attractors, bifurcations). In concern of motion of a body of elliptical profile in a viscous medium with imposed circulation of the velocity vector and with the applied constant torque, a presence of the Lorenztype strange attractor is discovered in the threedimensional space of generalized velocities.

Kuznetsov S. P., Kuznetsov A. S., Kruglov V. P.
Hyperbolic chaos in systems with parametrically excited patterns of standing waves
2014, Vol. 10, No. 3, pp. 265277
Abstract
We outline a possibility of implementation of Smale–Williams type attractors with different stretching factors for the angular coordinate, namely, $n=3,\,5,\,7,\,9,\,11$, for the maps describing the evolution of parametrically excited standing wave patterns on a nonlinear string over a period of modulation of pump accompanying by alternate excitation of modes with the wavelength ratios of $1:n$.

Isaeva O. B., Kuznetsov S. P., Sataev I. R., Pikovsky A.
On a bifurcation scenario of a birth of attractor of Smale–Williams type
2013, Vol. 9, No. 2, pp. 267294
Abstract
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 saddlenode 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 nonautonomous 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 A. S., Kuznetsov S. P.
Hyperbolic chaos in parametric oscillations of a string
2013, Vol. 9, No. 1, pp. 310
Abstract
We outline a possibility of chaotic dynamics associated with a hyperbolic attractor of the Smale–Williams type in mechanical vibrations of a nonhomogeneous string with nonlinear dissipation arising due to parametric excitation of modes at the frequencies $\omega$ and $3\omega$, when the pump is supplied by means of the string tension variations alternately at frequencies of $2\omega$ and $6\omega$.

Kuznetsov A. P., Turukina L. V., Kuznetsov S. P., Sataev I. R.
Landau–Hopf scenario in the ensemble of interacting oscillators
2012, Vol. 8, No. 5, pp. 863873
Abstract
The conditions are discussed for which the ensemble of interacting oscillators may demonstrate Landau–Hopf scenario of successive birth of multifrequency 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 quasiperiodic Hopf bifurcation.

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
2012, Vol. 8, No. 4, pp. 735762
Abstract
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 threedimensional 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 perioddoubling 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 A. P., Kuznetsov S. P., Pozdnyakov M. V., Sedova Y. V.
Universal twodimensional map and its radiophysical realization
2012, Vol. 8, No. 3, pp. 461471
Abstract
We suggest a simple twodimensional map, parameters of which are the trace and Jacobian of the perturbation matrix of the fixed point. On the parameters plane it demonstrates the main universal bifurcation scenarios: the threshold to chaos via perioddoublings, the situation of quasiperiodic oscillations and Arnold tongues. We demonstrate the possibility of implementation of such map in radiophysical device.

Kuznetsov S. P.
An example of a nonautonomous continuoustime system with attractor of Plykin type in the Poincare map
2009, Vol. 5, No. 3, pp. 403424
Abstract
A nonautonomous flow system is introduced, which may serve as a base for elaboration of real systems and devices demonstrating the structurally stable chaotic dynamics. The starting point is the map of the sphere composed of four stages of sequential continuous geometrically evident transformations. The computations indicate that in a certain parameter range the map posseses an attractor of Plykin type. Accounting the structural stability intrinsic to this attractor, a modification of the model is undertaken, which includes a variable change with passage to representation of instantaneous states on the plane. As a result, a set of two nonautonomous differential equations of the first order with smooth coefficients is obtained explicitly, which has the Plykin type attractor in the plane in the Poincaré crosssection. Results of computations are presented for the sphere map and for the flow system including portraits of attractors, Lyapunov exponents, dimension estimates. Substantiation of the hyperbolic nature of the attractors for the sphere map and for the flow system is based on a computer procedure of verification of the socalled cone criterion; in this context, some hints are applied, which may be useful in similar computations for some other systems.

Kuznetsov A. P., Kuznetsov S. P., Sataev I. R., Sedova Y. V.
Critical point of accumulation of foldflip bifurcation points and critical quasiattractor (the review and new results)
2008, Vol. 4, No. 2, pp. 113132
Abstract
In paper we suggest an example of system which dynamics is answered to conception of a «critical quasiattractor». 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 quasiattractor, 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.

Kuptsov P. V., Kuznetsov S. P.
Transition to a synchronous chaos regime in a system of coupled nonautonomous oscillators presented in terms of amplitude equations
2006, Vol. 2, No. 3, pp. 307331
Abstract
Amplitude equations are obtained for a system of two coupled van der Pol oscillators that has been recently suggested as a simple system with hyperbolic chaotic attractor allowing physical realization. We demonstrate that an approximate model based on the amplitude equations preserves basic features of a hyperbolic dynamics of the initial system. For two coupled amplitude equations models having the hyperbolic attractors a transition to synchronous chaos is studied. Phenomena typically accompanying this transition, as riddling and bubbling, are shown to manifest themselves in a specific way and can be observed only in a small vicinity of a critical point. Also, a structure of manydimensional attractor of the system is described in a region below the synchronization point.
