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2013
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# Sergey Kuznetsov

ul. Zelenaya 38, Saratov, 410019, Russia

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.

## 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:

 Kruglov V. P., Kuznetsov S. P. Topaj – Pikovsky Involution in the Hamiltonian Lattice of Locally Coupled Oscillators 2019, vol. 24, no. 6, pp.  725-738 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. Keywords: reversibility, involution, Hamiltonian system, Topaj – Pikovsky model, phase oscillator lattice Citation: Kruglov V. P., Kuznetsov S. P.,  Topaj – Pikovsky Involution in the Hamiltonian Lattice of Locally Coupled Oscillators, Regular and Chaotic Dynamics, 2019, vol. 24, no. 6, pp. 725-738 DOI:10.1134/S1560354719060108
 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. 7-8, pp.  803-820 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 two-dimensional 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. Keywords: Chaplygin sleigh, friction, parametric oscillator, strange attractor, Lyapunov exponents, chaotic dynamics, fish-like robot Citation: 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, Regular and Chaotic Dynamics, 2018, vol. 23, no. 7-8, pp. 803-820 DOI:10.1134/S1560354718070018
 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. 7-8, pp.  908-932 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 well-known 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 finite-time 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 finite-time 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. Keywords: chaotic attractor, strange pseudohyperbolic attractor, method of angles, hyperbolic isolation, Lyapunov exponents, finite-time Lyapunov exponents, instant Lyapunov exponents, covariant Lyapunov vectors Citation: Kuptsov P. V., Kuznetsov S. P.,  Lyapunov Analysis of Strange Pseudohyperbolic Attractors: Angles Between Tangent Subspaces, Local Volume Expansion and Contraction, Regular and Chaotic Dynamics, 2018, vol. 23, no. 7-8, pp. 908-932 DOI:10.1134/S1560354718070079
 Kuznetsov S. P., Sedova Y. V. Hyperbolic Chaos in Systems Based on FitzHugh–Nagumo Model Neurons 2018, vol. 23, no. 4, pp.  458-470 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 time-delay feedback loop with a quadratic nonlinear element. Keywords: hyperbolic chaos, Smale–Williams solenoid, FitzHugh–Nagumo neuron, time-delay system Citation: Kuznetsov S. P., Sedova Y. V.,  Hyperbolic Chaos in Systems Based on FitzHugh–Nagumo Model Neurons, Regular and Chaotic Dynamics, 2018, vol. 23, no. 4, pp. 458-470 DOI:10.1134/S1560354718040068
 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.  178-192 Abstract The main goal of the article is to suggest a two-dimensional map that could play the role of a generalized model similar to the standard Chirikov–Taylor mapping, but appropriate for energy-conserving 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 quasi-periodic regimes take place similar to those observed in conservative nonlinear dynamics. Keywords: nonholonomic mechanics, Chaplygin sleigh, attractor, chaos, bifurcation, Chirikov–Taylor map Citation: Kuznetsov S. P.,  Regular and Chaotic Dynamics of a Chaplygin Sleigh due to Periodic Switch of the Nonholonomic Constraint, Regular and Chaotic Dynamics, 2018, vol. 23, no. 2, pp. 178-192 DOI:10.1134/S1560354718020041
 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