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:
Kruglov V. P., Kuznetsov S. P.
Topaj – Pikovsky Involution in the Hamiltonian Lattice of Locally Coupled Oscillators
2019, vol. 24, no. 6, pp. 725738
Abstract
We discuss the Hamiltonian model of an oscillator lattice with local coupling. The
Hamiltonian model describes localized spatial modes of nonlinear the Schrödinger equation
with periodic tilted potential. The Hamiltonian system manifests reversibility of the Topaj –
Pikovsky phase oscillator lattice. Furthermore, the Hamiltonian system has invariant manifolds
with asymptotic dynamics exactly equivalent to the Topaj – Pikovsky model. We examine the
stability of trajectories belonging to invariant manifolds by means of numerical evaluation of
Lyapunov exponents. We show that there is no contradiction between asymptotic dynamics
on invariant manifolds and conservation of phase volume of the Hamiltonian system. We
demonstrate the complexity of dynamics with results of numerical simulations.

Borisov A. V., Kuznetsov S. P.
Comparing Dynamics Initiated by an Attached Oscillating Particle for the Nonholonomic Model of a Chaplygin Sleigh and for a Model with Strong Transverse and Weak Longitudinal Viscous Friction Applied at a Fixed Point on the Body
2018, vol. 23, no. 78, pp. 803820
Abstract
This paper addresses the problem of a rigid body moving on a plane (a platform) whose motion is initiated by oscillations of a point mass relative to the body in the presence of the viscous friction force applied at a fixed point of the platform and having in one direction a small (or even zero) value and a large value in the transverse direction. This problem is analogous to that of a Chaplygin sleigh when the nonholonomic constraint prohibiting motions of the fixed point on the platform across the direction prescribed on it is replaced by viscous friction. We present numerical results which confirm correspondence between the phenomenology of complex dynamics of the model with a nonholonomic constraint and a system with viscous friction —
phase portraits of attractors, bifurcation diagram, and Lyapunov exponents. In particular, we show the possibility of the platform’s motion being accelerated by oscillations of the internal mass, although, in contrast to the nonholonomic model, the effect of acceleration tends to saturation. We also show the possibility of chaotic dynamics related to strange attractors of equations for generalized velocities, which is accompanied by a twodimensional random walk of the platform in a laboratory reference system. The results obtained may be of interest to applications in the context of the problem of developing robotic mechanisms for motion in a fluid under the action of the motions of internal masses.

Kuptsov P. V., Kuznetsov S. P.
Lyapunov Analysis of Strange Pseudohyperbolic Attractors: Angles Between Tangent Subspaces, Local Volume Expansion and Contraction
2018, vol. 23, no. 78, pp. 908932
Abstract
Pseudohyperbolic attractors are genuine strange chaotic attractors. They do not contain stable periodic orbits and are robust in the sense that such orbits do not appear under variations. The tangent space of these attractors is split into a direct sum of volume expanding and contracting subspaces and these subspaces never have tangencies with each other. Any contraction in the first subspace, if it occurs, is weaker than contractions in the second one. In this paper we analyze the local structure of several chaotic attractors recently suggested in the literature as pseudohyperbolic. The absence of tangencies and thus the presence of the pseudohyperbolicity is verified using the method of angles that includes computation of distributions of the angles between the corresponding tangent subspaces. Also, we analyze how volume expansion in the first subspace and the contraction in the second one occurs locally. For this purpose we introduce a family of instant Lyapunov exponents. Unlike the wellknown
finite time ones, the instant Lyapunov exponents show expansion or contraction on infinitesimal time intervals. Two types of instant Lyapunov exponents are defined. One is related to ordinary finitetime Lyapunov exponents computed in the course of standard algorithm for Lyapunov exponents. Their sums reveal instant volume expanding properties. The second type of instant
Lyapunov exponents shows how covariant Lyapunov vectors grow or decay on infinitesimal time. Using both instant and finitetime Lyapunov exponents, we demonstrate that average expanding and contracting properties specific to pseudohyperbolicity are typically violated on infinitesimal time. Instantly volumes from the first subspace can sometimes be contacted, directions in the
second subspace can sometimes be expanded, and the instant contraction in the first subspace can sometimes be stronger than the contraction in the second subspace.

Kuznetsov S. P., Sedova Y. V.
Hyperbolic Chaos in Systems Based on FitzHugh–Nagumo Model Neurons
2018, vol. 23, no. 4, pp. 458470
Abstract
In the present paper we consider and study numerically two systems based on model FitzHugh–Nagumo neurons, where in the presence of periodic modulation of parameters it is possible to implement chaotic dynamics on the attractor in the form of a Smale–Williams solenoid in the stroboscopic Poincaré map. In particular, hyperbolic chaos characterized by structural stability occurs in a single neuron supplemented by a timedelay feedback loop with a quadratic nonlinear element.

Kuznetsov S. P.
Regular and Chaotic Dynamics of a Chaplygin Sleigh due to Periodic Switch of the Nonholonomic Constraint
2018, vol. 23, no. 2, pp. 178192
Abstract
The main goal of the article is to suggest a twodimensional map that could play the role of a generalized model similar to the standard Chirikov–Taylor mapping, but appropriate for energyconserving nonholonomic dynamics. In this connection, we consider a Chaplygin sleigh on a plane, supposing that the nonholonomic constraint switches periodically in such a way that it is located alternately at each of three legs supporting the sleigh. We assume that at the initiation of the constraint the respective element (“knife edge”) is directed along the local velocity vector and becomes instantly fixed relative to the sleigh till the next switch. Differential equations of the mathematical model are formulated and an analytical derivation of mapping for the state evolution on the switching period is provided. The dynamics take place with conservation of the mechanical energy, which plays the role of one of the parameters responsible for the type of the dynamic behavior. At the same time, the Liouville theorem does not hold, and the phase volume can undergo compression or expansion in certain state space domains. Numerical simulations reveal phenomena characteristic of nonholonomic systems with complex dynamics (like the rattleback or the Chaplygin top). In particular, on the energy surface attractors associated with regular sustained motions can occur, settling in domains of prevalent phase volume compression together with repellers in domains of the phase volume expansion. In addition, chaotic and quasiperiodic regimes take place similar to those observed in conservative nonlinear dynamics.

Kuptsov P. V., Kuznetsov S. P., Stankevich N. V.
A Family of Models with Blue Sky Catastrophes of Different Classes
2017, vol. 22, no. 5, pp. 551565
Abstract
A generalized model with bifurcations associated with blue sky catastrophes is introduced. Depending on an integer index $m$, different kinds of attractors arise, including those associated with quasiperiodic oscillations and with hyperbolic chaos. Verification of the hyperbolicity is provided based on statistical analysis of intersection angles of stable and unstable manifolds.

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.

Borisov A. V., Kuznetsov S. P.
Regular and Chaotic Motions of a Chaplygin Sleigh under Periodic Pulsed Torque Impacts
2016, vol. 21, no. 78, pp. 792803
Abstract
For a Chaplygin sleigh on a plane, which is a paradigmatic system of nonholonomic mechanics, we consider dynamics driven by periodic pulses of supplied torque depending on the instant spatial orientation of the sleigh. Additionally, we assume that a weak viscous force and moment affect the sleigh in time intervals between the pulses to provide sustained modes of the motion associated with attractors in the reduced threedimensional phase space (velocity, angular velocity, rotation angle). The developed discrete version of the problem of the Chaplygin sleigh is an analog of the classical Chirikov map appropriate for the nonholonomic situation. We demonstrate numerically, discuss and classify dynamical regimes depending on the parameters, including regular motions and diffusivelike random walks associated, respectively, with regular and chaotic attractors in the reduced momentum dynamical equations.

Kuznetsov S. P., Kruglov V. P.
Verification of Hyperbolicity for Attractors of Some Mechanical Systems with Chaotic Dynamics
2016, vol. 21, no. 2, pp. 160174
Abstract
Computer verification of hyperbolicity is provided based on statistical analysis of the angles of intersection of stable and unstable manifolds for mechanical systems with hyperbolic attractors of Smale – Williams type: (i) a particle sliding on a plane under periodic kicks, (ii) interacting particles moving on two alternately rotating disks, and (iii) a string with parametric excitation of standingwave patterns by a modulated pump. The examples are of interest as contributing to filling the hyperbolic theory of dynamical systems with physical content.

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
2015, vol. 20, no. 6, pp. 649666
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.
Plate Falling in a Fluid: Regular and Chaotic Dynamics of Finitedimensional Models
2015, vol. 20, no. 3, pp. 345382
Abstract
Results are reviewed concerning the planar problem of a plate falling in a resisting medium studied with models based on ordinary differential equations for a small number of dynamical variables. A unified model is introduced to conduct a comparative analysis of the dynamical 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 manifests certain similarities caused by the same inherent symmetry and by the universal nature of the phenomena involved in nonlinear dynamics (fixed points, limit cycles, attractors, and bifurcations).

Kruglov V. P., Kuznetsov S. P., Pikovsky A.
Attractor of Smale–Williams Type in an Autonomous Distributed System
2014, vol. 19, no. 4, pp. 483494
Abstract
We consider an autonomous system of partial differential equations for a onedimensional distributed medium with periodic boundary conditions. Dynamics in time consists of alternating birth and death of patterns with spatial phases transformed from one stage of activity to another by the doubly expanding circle map. So, the attractor in the Poincaré section is uniformly hyperbolic, a kind of Smale–Williams solenoid. Finitedimensional models are derived as ordinary differential equations for amplitudes of spatial Fourier modes (the 5D and 7D models). Correspondence of the reduced models to the original system is demonstrated numerically. Computational verification of the hyperbolicity criterion is performed for the reduced models: the distribution of angles of intersection for stable and unstable manifolds on the attractor is separated from zero, i.e., the touches are excluded. The example considered gives a partial justification for the old hopes that the chaotic behavior of autonomous distributed systems may be associated with uniformly hyperbolic attractors.

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.

Kuznetsov S. P.
Example of blue sky catastrophe accompanied by a birth of Smale–Williams attractor
2010, vol. 15, no. 23, pp. 348353
Abstract
A model system is proposed, which manifests a blue sky catastrophe giving rise to a hyperbolic attractor of Smale–Williams type in accordance with theory of Shilnikov and Turaev. Some essential features of the transition are demonstrated in computations, including Bernoullitype discretestep evolution of the angular variable, inverse square root dependence of the first return time on the bifurcation parameter, certain type of dependence of Lyapunov exponents on control parameter for the differential equations and for the Poincaré map.

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.

Kuznetsov S. P.
Generalized dimensions of the goldenmean quasiperiodic orbit from renormalizationgroup functional equation
2005, vol. 10, no. 1, pp. 3338
Abstract
A method is suggested for computation of the generalized dimensions for a fractal attractor associated with the quasiperiodic transition to chaos at the goldenmean rotation number. The approach is based on an eigenvalue problem formulated in terms of functional equations with coeficients expressed via the universal fixedpoint function of Feigenbaum–Kadanoff–Shenker. The accuracy of the results is determined only by precision of representation of the universal function.

Kuznetsov S. P., Osbaldestin A. H.
Generalized Dimensions of Feigenbaum's Attractor from RenormalizationGroup Functional Equations
2002, vol. 7, no. 3, pp. 325330
Abstract
A method is suggested for the computation of the generalized dimensions of fractal attractors at the perioddoubling transition to chaos. The approach is based on an eigenvalue problem formulated in terms of functional equations with coefficients expressed in terms of Feigenbaum's universal fixedpoint function. The accuracy of the results depends only on the accuracy of the representation of the universal function.

Isaeva O. B., Kuznetsov S. P.
On scaling properties of twodimensional maps near the accumulation point of the periodtripling cascade
2000, vol. 5, no. 4, pp. 459476
Abstract
We analyse dynamics generated by quadratic complex map at the accumulation point of the periodtripling cascade (see Golberg, Sinai, and Khanin, Usp. Mat. Nauk. V. 38, № 1, 1983, 159; Cvitanovic; and Myrheim, Phys. Lett. A94, № 8, 1983, 329). It is shown that in general this kind of the universal behavior does not survive the translation twodimensional real maps violating the Cauchy–Riemann equations. In the extended parameter space of the twodimensional maps the scaling properties are determined by two complex universal constants. One of them corresponds to perturbations retaining the map in the complexanalytic class and equals $\delta_1 \cong 4.60028.9812i$ in accordance with the mentioned works. The second constant $\delta_2 \cong 2.5872+1.8067i$ is responsible for violation of the analyticity. Graphical illustrations of scaling properties associated with both these constants are presented. We conclude that in the extended parameter space of the twodimensional maps the period tripling universal behavior appears as a phenomenon of codimension $4$.

Kuznetsov A. P., Kuznetsov S. P., Sataev I. R.
Codimension and typicity in a context of description of transition to chaos via perioddoubling in dissipative dynamical systems
1997, vol. 2, nos. 34, pp. 90105
Abstract
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 onedimensional maps and discuss a possibility of their realization and observation in nonlinear dissipative systems of more general form.
