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Sergey Kuznetsov

Sergey Kuznetsov
ul. Zelenaya 38, Saratov, 410019, Russia
Kotelnikov’s Institute of Radio-Engineering and Electronics, Russian Academy of Sciences

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 Physical-Technical 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 1974-1977 he is a post-graduate 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 non-stationary phenomena in certain extended self-oscillating systems of interacting electron beam and electromagnetic wave", speciality Radio-physics. 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: "Non-stationary nonlinear processes and stochastic oscillations in spatially extended systems of radio-physics and electronics". From 1988 S.P.Kuznetsov is a Head Researcher of Saratov Branch ofKotel'nikov's Institute of Radio-Engineering and Electronics of RAS. In parallel, in 1992 - 1995 he is a Professor of Chair of Radio-physics 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:

1994-1996: S.P.Kuznetsov was a Laureate of State Stipendium for Distinguished Scientists of Russian Federation
1988: Soros Associated Professor
2000, 2001: Soros Professor

Publications:

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.  551-565
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 quasi-periodic oscillations and with hyperbolic chaos. Verification of the hyperbolicity is provided based on statistical analysis of intersection angles of stable and unstable manifolds.
Keywords: dynamical system, blue sky catastrophe, quasi-periodic oscillations, hyperbolic chaos, Smale–Williams solenoid
Citation: Kuptsov P. V., Kuznetsov S. P., Stankevich N. V.,  A Family of Models with Blue Sky Catastrophes of Different Classes, Regular and Chaotic Dynamics, 2017, vol. 22, no. 5, pp. 551-565
DOI:10.1134/S1560354717050069
Jalnine A. Y., Kuznetsov S. P.
Autonomous Strange Nonchaotic Oscillations in a System of Mechanical Rotators
2017, vol. 22, no. 3, pp.  210-225
Abstract
We investigate strange nonchaotic self-oscillations 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 vibrational-rotational 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 three-frequency quasi-periodic, chaotic and strange nonchaotic self-oscillations. 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.
Keywords: autonomous dynamical system, mechanical rotators, quasi-periodic oscillations, strange nonchaotic attractor, chaos
Citation: Jalnine A. Y., Kuznetsov S. P.,  Autonomous Strange Nonchaotic Oscillations in a System of Mechanical Rotators, Regular and Chaotic Dynamics, 2017, vol. 22, no. 3, pp. 210-225
DOI:10.1134/S1560354717030029
Borisov A. V., Kuznetsov S. P.
Regular and Chaotic Motions of a Chaplygin Sleigh under Periodic Pulsed Torque Impacts
2016, vol. 21, no. 7-8, pp.  792-803
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 three-dimensional 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 diffusive-like random walks associated, respectively, with regular and chaotic attractors in the reduced momentum dynamical equations.
Keywords: Chaplygin sleigh, nonholonomic mechanics, attractor, chaos, bifurcation
Citation: Borisov A. V., Kuznetsov S. P.,  Regular and Chaotic Motions of a Chaplygin Sleigh under Periodic Pulsed Torque Impacts, Regular and Chaotic Dynamics, 2016, vol. 21, no. 7-8, pp. 792-803
DOI:10.1134/S1560354716070029
Kuznetsov S. P., Kruglov V. P.
Verification of Hyperbolicity for Attractors of Some Mechanical Systems with Chaotic Dynamics
2016, vol. 21, no. 2, pp.  160-174
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 standing-wave patterns by a modulated pump. The examples are of interest as contributing to filling the hyperbolic theory of dynamical systems with physical content.
Keywords: dynamical system, chaos, attractor, hyperbolic dynamics, Lyapunov exponent, Smale – Williams solenoid, parametric oscillations
Citation: Kuznetsov S. P., Kruglov V. P.,  Verification of Hyperbolicity for Attractors of Some Mechanical Systems with Chaotic Dynamics, Regular and Chaotic Dynamics, 2016, vol. 21, no. 2, pp. 160-174
DOI:10.1134/S1560354716020027
Kuznetsov S. P.
Hyperbolic Chaos in Self-oscillating Systems Based on Mechanical Triple Linkage: Testing Absence of Tangencies of Stable and Unstable Manifolds for Phase Trajectories
2015, vol. 20, no. 6, pp.  649-666
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 self-oscillating 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.
Keywords: dynamical system, chaos, hyperbolic attractor, Anosov dynamics, rotator, Lyapunov exponent, self-oscillator
Citation: Kuznetsov S. P.,  Hyperbolic Chaos in Self-oscillating Systems Based on Mechanical Triple Linkage: Testing Absence of Tangencies of Stable and Unstable Manifolds for Phase Trajectories, Regular and Chaotic Dynamics, 2015, vol. 20, no. 6, pp. 649-666
DOI:10.1134/S1560354715060027
Kuznetsov S. P.
Plate Falling in a Fluid: Regular and Chaotic Dynamics of Finite-dimensional Models
2015, vol. 20, no. 3, pp.  345-382
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).
Keywords: body motion in a fluid, oscillations, autorotation, flutter, attractor, bifurcation, chaos, Lyapunov exponent
Citation: Kuznetsov S. P.,  Plate Falling in a Fluid: Regular and Chaotic Dynamics of Finite-dimensional Models, Regular and Chaotic Dynamics, 2015, vol. 20, no. 3, pp. 345-382
DOI:10.1134/S1560354715030090
Kruglov V. P., Kuznetsov S. P., Pikovsky A.
Attractor of Smale–Williams Type in an Autonomous Distributed System
2014, vol. 19, no. 4, pp.  483-494
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. Finite-dimensional 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.
Keywords: Smale–Williams solenoid, hyperbolic attractor, chaos, Swift–Hohenberg equation, Lyapunov exponent
Citation: Kruglov V. P., Kuznetsov S. P., Pikovsky A.,  Attractor of Smale–Williams Type in an Autonomous Distributed System, Regular and Chaotic Dynamics, 2014, vol. 19, no. 4, pp. 483-494
DOI:10.1134/S1560354714040042
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.  512-532
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 quasi-conservative 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 period-doubling 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.
Keywords: rattleback, rigid body dynamics, nonholonomic mechanics, strange attractor, Lyapunov exponents, bifurcation, fractal dimension
Citation: 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, Regular and Chaotic Dynamics, 2012, vol. 17, no. 6, pp. 512-532
DOI:10.1134/S1560354712060044
Kuznetsov S. P.
Example of blue sky catastrophe accompanied by a birth of Smale–Williams attractor
2010, vol. 15, no. 2-3, pp.  348-353
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 Bernoulli-type discrete-step 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.
Keywords: attractor, bifurcation, Smale–Williams solenoid, Lyapunov exponent
Citation: Kuznetsov S. P.,  Example of blue sky catastrophe accompanied by a birth of Smale–Williams attractor, Regular and Chaotic Dynamics, 2010, vol. 15, no. 2-3, pp. 348-353
DOI:10.1134/S1560354710020206
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.  19-30
Abstract
We consider the dynamics of small perturbations of stable two-frequency 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 three-frequency 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.
Keywords: quasiperiodicity, strange nonchaotic attractor, bifurcation, stability analysis
Citation: 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 , Regular and Chaotic Dynamics, 2006, vol. 11, no. 1, pp. 19-30
DOI:10.1070/RD2006v011n01ABEH000332
Kuznetsov S. P.
Generalized dimensions of the golden-mean quasiperiodic orbit from renormalization-group functional equation
2005, vol. 10, no. 1, pp.  33-38
Abstract
A method is suggested for computation of the generalized dimensions for a fractal attractor associated with the quasiperiodic transition to chaos at the golden-mean rotation number. The approach is based on an eigenvalue problem formulated in terms of functional equations with coeficients expressed via the universal fixed-point function of Feigenbaum–Kadanoff–Shenker. The accuracy of the results is determined only by precision of representation of the universal function.
Keywords: circle map, golden mean, renormalization, dimension, generalized dimensions
Citation: Kuznetsov S. P.,  Generalized dimensions of the golden-mean quasiperiodic orbit from renormalization-group functional equation, Regular and Chaotic Dynamics, 2005, vol. 10, no. 1, pp. 33-38
DOI:10.1070/RD2005v010n01ABEH000298
Kuznetsov S. P., Osbaldestin A. H.
Generalized Dimensions of Feigenbaum's Attractor from Renormalization-Group Functional Equations
2002, vol. 7, no. 3, pp.  325-330
Abstract
A method is suggested for the computation of the generalized dimensions of fractal attractors at the period-doubling 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 fixed-point function. The accuracy of the results depends only on the accuracy of the representation of the universal function.
Citation: Kuznetsov S. P., Osbaldestin A. H.,  Generalized Dimensions of Feigenbaum's Attractor from Renormalization-Group Functional Equations, Regular and Chaotic Dynamics, 2002, vol. 7, no. 3, pp. 325-330
DOI:10.1070/RD2002v007n03ABEH000214
Isaeva O. B., Kuznetsov S. P.
On scaling properties of two-dimensional maps near the accumulation point of the period-tripling cascade
2000, vol. 5, no. 4, pp.  459-476
Abstract
We analyse dynamics generated by quadratic complex map at the accumulation point of the period-tripling 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 two-dimensional real maps violating the Cauchy–Riemann equations. In the extended parameter space of the two-dimensional maps the scaling properties are determined by two complex universal constants. One of them corresponds to perturbations retaining the map in the complex-analytic class and equals $\delta_1 \cong 4.6002-8.9812i$ 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 two-dimensional maps the period tripling universal behavior appears as a phenomenon of codimension $4$.
Citation: Isaeva O. B., Kuznetsov S. P.,  On scaling properties of two-dimensional maps near the accumulation point of the period-tripling cascade, Regular and Chaotic Dynamics, 2000, vol. 5, no. 4, pp. 459-476
DOI:10.1070/RD2000v005n04ABEH000159
Kuznetsov A. P., Kuznetsov S. P., Sataev I. R.
Codimension and typicity in a context of description of transition to chaos via period-doubling in dissipative dynamical systems
1997, vol. 2, nos. 3-4, pp.  90-105
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 one-dimensional maps and discuss a possibility of their realization and observation in nonlinear dissipative systems of more general form.
Citation: Kuznetsov A. P., Kuznetsov S. P., Sataev I. R.,  Codimension and typicity in a context of description of transition to chaos via period-doubling in dissipative dynamical systems, Regular and Chaotic Dynamics, 1997, vol. 2, nos. 3-4, pp. 90-105
DOI:10.1070/RD1997v002n04ABEH000050

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