A Family of Models with Blue Sky Catastrophes of Different Classes
Regular and Chaotic Dynamics, 2017, vol. 22, no. 5, pp. 551-565
Abstract
pdf (1.61 Mb)
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.
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
Autonomous Strange Nonchaotic Oscillations in a System of Mechanical Rotators
Regular and Chaotic Dynamics, 2017, vol. 22, no. 3, pp. 210-225
Abstract
pdf (2.26 Mb)
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.
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
Chaos generator with the Smale–Williams attractor based on oscillation death
Russian Journal of Nonlinear Dynamics, 2017, vol. 13, no. 3, pp. 303-315
Abstract
pdf (5.99 Mb)
A nonautonomous system with a uniformly hyperbolic attractor of Smale – Williams type in
a Poincaré cross-section 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 cross-section, 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.
Doroshenko V. M., Kruglov V. P., Kuznetsov S. P., Chaos generator with the Smale–Williams attractor based on oscillation death, Russian Journal of Nonlinear Dynamics, 2017, vol. 13, no. 3, pp. 303-315
Autonomous strange non-chaotic oscillations in a system of mechanical rotators
Russian Journal of Nonlinear Dynamics, 2017, vol. 13, no. 2, pp. 257-275
Abstract
pdf (907.9 Kb)
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.
Jalnine A. Y., Kuznetsov S. P., Autonomous strange non-chaotic oscillations in a system of mechanical rotators, Russian Journal of Nonlinear Dynamics, 2017, vol. 13, no. 2, pp. 257-275
Describing the Motion of a Body with an Elliptical Cross Section in a Viscous Uncompressible Fluid by Model Equations Reconstructed from Data Processing
Technical Physics Letters, 2016, vol. 42, no. 9, pp. 886-890
Abstract
pdf (508.12 Kb)
From analysis of time series obtained on the numerical solution of a plane problem on the motion
of a body with an elliptic cross section under the action of gravity force in an incompressible viscous fluid, a
system of ordinary differential equations approximately describing the dynamics of the body is reconstructed.
To this end, coefficients responsible for the added mass, the force caused by the circulation of the velocity
field, and the resisting force are found by the least square adjustment. The agreement between the finitedimensional
description and the simulation on the basis of the Navier–Stokes equations is illustrated by
images of attractors in regular and chaotic modes. The coefficients found make it possible to estimate the
actual contribution of different effects to the dynamics of the body.
Citation:
Borisov A. V., Kuznetsov S. P., Mamaev I. S., Tenenev V. A., Describing the Motion of a Body with an Elliptical Cross Section in a Viscous Uncompressible Fluid by Model Equations Reconstructed from Data Processing, Technical Physics Letters, 2016, vol. 42, no. 9, pp. 886-890
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
Abstract
pdf (1.65 Mb)
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.
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
Verification of Hyperbolicity for Attractors of Some Mechanical Systems with Chaotic Dynamics
Regular and Chaotic Dynamics, 2016, vol. 21, no. 2, pp. 160-174
Abstract
pdf (1.14 Mb)
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.
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
Pendulum system with an infinite number of equilibrium states and quasiperiodic dynamics
Russian Journal of Nonlinear Dynamics, 2016, vol. 12, no. 2, pp. 223-234
Abstract
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Examples of mechanical systems are discussed, where quasi-periodic 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 self-oscillating system the presence of many attractors in the form of limit cycles and of quasi-periodic rotational modes is observed. In the case of quasi-periodic 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 A. P., Kuznetsov S. P., Sedova Y. V., Pendulum system with an infinite number of equilibrium states and quasiperiodic dynamics, Russian Journal of Nonlinear Dynamics, 2016, vol. 12, no. 2, pp. 223-234
Hyperbolic chaos in self-oscillating systems based on mechanical triple linkage: Testing absence of tangencies of stable and unstable manifolds for phase trajectories
Russian Journal of Nonlinear Dynamics, 2016, vol. 12, no. 1, pp. 121-143
Abstract
pdf (1.96 Mb)
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.
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, Russian Journal of Nonlinear Dynamics, 2016, vol. 12, no. 1, pp. 121-143
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
Abstract
pdf (3.15 Mb)
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.
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
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
Abstract
pdf (1.5 Mb)
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
Motion of a falling card in a fluid: Finite-dimensional models, complex phenomena, and nonlinear dynamics
Russian Journal of Nonlinear Dynamics, 2015, vol. 11, no. 1, pp. 3-49
Abstract
pdf (1.39 Mb)
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 Lorenz-type strange attractor is discovered in the three-dimensional space of generalized velocities.
Keywords:
body motion in fluid, oscillations, autorotation, flutter, attractor, bifurcation, chaos, Lyapunov exponent
Citation:
Kuznetsov S. P., Motion of a falling card in a fluid: Finite-dimensional models, complex phenomena, and nonlinear dynamics, Russian Journal of Nonlinear Dynamics, 2015, vol. 11, no. 1, pp. 3-49
Nonlinear dynamics of the rattleback: a nonholonomic model
Physics-Uspekhi, 2014, vol. 184, no. 5, pp. 453-460
Abstract
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For a solid body of convex form moving on a rough horizontal plane that is known as a rattleback, numerical simulations are used to discuss and illustrate dynamical phenomena that are characteristic of the motion due to a nonholonomic nature of the mechanical system; the relevant feature is the nonconservation of the phase volume in the course of the dynamics. In such a system, a local compression of the phase volume can produce behavior features similar to those exhibited by dissipative systems, such as stable equilibrium points corresponding to stationary rotations; limit cycles (rotations with oscillations); and strange attractors. A chart of dynamical regimes is plotted in a plane whose axes are the total mechanical energy and the relative angle between the geometric and dynamic principal axes of the body. The transition to chaos through a sequence of Feigenbaum period doubling bifurcations is demonstrated. A number of strange attractors are considered, for which phase portraits, Lyapunov exponents, and Fourier spectra are presented.
Citation:
Borisov A. V., Kazakov A. O., Kuznetsov S. P., Nonlinear dynamics of the rattleback: a nonholonomic model , Physics-Uspekhi, 2014, vol. 184, no. 5, pp. 453-460
Attractor of Smale–Williams Type in an Autonomous Distributed System
Regular and Chaotic Dynamics, 2014, vol. 19, no. 4, pp. 483-494
Abstract
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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.
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
Hyperbolic chaos in systems with parametrically excited patterns of standing waves
Russian Journal of Nonlinear Dynamics, 2014, vol. 10, no. 3, pp. 265-277
Abstract
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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 $1:n$.
Kuznetsov S. P., Kuznetsov A. S., Kruglov V. P., Hyperbolic chaos in systems with parametrically excited patterns of standing waves, Russian Journal of Nonlinear Dynamics, 2014, vol. 10, no. 3, pp. 265-277
On a bifurcation scenario of a birth of attractor of Smale–Williams type
Russian Journal of Nonlinear Dynamics, 2013, vol. 9, no. 2, pp. 267-294
Abstract
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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 saddle-node 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 non-autonomous 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 S. P., Sataev I. R., Pikovsky A., On a bifurcation scenario of a birth of attractor of Smale–Williams type, Russian Journal of Nonlinear Dynamics, 2013, vol. 9, no. 2, pp. 267-294
Hyperbolic chaos in parametric oscillations of a string
Russian Journal of Nonlinear Dynamics, 2013, vol. 9, no. 1, pp. 3-10
Abstract
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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 $6\omega$.
Isaeva O. B., Kuznetsov A. S., Kuznetsov S. P., Hyperbolic chaos in parametric oscillations of a string, Russian Journal of Nonlinear Dynamics, 2013, vol. 9, no. 1, pp. 3-10
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
Abstract
pdf (4.23 Mb)
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.
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
Landau–Hopf scenario in the ensemble of interacting oscillators
Russian Journal of Nonlinear Dynamics, 2012, vol. 8, no. 5, pp. 863-873
Abstract
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The conditions are discussed for which the ensemble of interacting oscillators may demonstrate Landau–Hopf scenario of successive birth of multi-frequency 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 quasi-periodic Hopf bifurcation.
Kuznetsov A. P., Turukina L. V., Kuznetsov S. P., Sataev I. R., Landau–Hopf scenario in the ensemble of interacting oscillators, Russian Journal of Nonlinear Dynamics, 2012, vol. 8, no. 5, pp. 863-873
Phenomena of nonlinear dynamics of dissipative systems in nonholonomic mechanics of the rattleback
Russian Journal of Nonlinear Dynamics, 2012, vol. 8, no. 4, pp. 735-762
Abstract
pdf (1.42 Mb)
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 three-dimensional 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 period-doubling 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 S. P., Jalnine A. Y., Sataev I. R., Sedova Y. V., Phenomena of nonlinear dynamics of dissipative systems in nonholonomic mechanics of the rattleback, Russian Journal of Nonlinear Dynamics, 2012, vol. 8, no. 4, pp. 735-762
Universal two-dimensional map and its radiophysical realization
Russian Journal of Nonlinear Dynamics, 2012, vol. 8, no. 3, pp. 461-471
Abstract
pdf (637.32 Kb)
We suggest a simple two-dimensional 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 period-doublings, the situation of quasiperiodic oscillations and Arnold tongues. We demonstrate the possibility of implementation of such map in radiophysical device.
Keywords:
maps, bifurcations, phenomena of quasiperiodicity
Citation:
Kuznetsov A. P., Kuznetsov S. P., Pozdnyakov M. V., Sedova Y. V., Universal two-dimensional map and its radiophysical realization, Russian Journal of Nonlinear Dynamics, 2012, vol. 8, no. 3, pp. 461-471
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
Abstract
pdf (272.33 Kb)
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.
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
An example of a non-autonomous continuous-time system with attractor of Plykin type in the Poincare map
Russian Journal of Nonlinear Dynamics, 2009, vol. 5, no. 3, pp. 403-424
Abstract
pdf (2.26 Mb)
A non-autonomous 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 non-autonomous differential equations of the first order with smooth coefficients is obtained explicitly, which has the Plykin type attractor in the plane in the Poincaré cross-section. 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 so-called cone criterion; in this context, some hints are applied, which may be useful in similar computations for some other systems.
Kuznetsov S. P., An example of a non-autonomous continuous-time system with attractor of Plykin type in the Poincare map, Russian Journal of Nonlinear Dynamics, 2009, vol. 5, no. 3, pp. 403-424
Critical point of accumulation of fold-flip bifurcation points and critical quasi-attractor (the review and new results)
Russian Journal of Nonlinear Dynamics, 2008, vol. 4, no. 2, pp. 113-132
Abstract
pdf (1.02 Mb)
In paper we suggest an example of system which dynamics is answered to conception of a «critical quasi-attractor». 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 quasi-attractor, 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.
Keywords:
quasi-attractor, renormalization group method, type of criticality, bifurcation, scaling, noise
Citation:
Kuznetsov A. P., Kuznetsov S. P., Sataev I. R., Sedova Y. V., Critical point of accumulation of fold-flip bifurcation points and critical quasi-attractor (the review and new results), Russian Journal of Nonlinear Dynamics, 2008, vol. 4, no. 2, pp. 113-132
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
Abstract
pdf (520.95 Kb)
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.
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
Transition to a synchronous chaos regime in a system of coupled non-autonomous oscillators presented in terms of amplitude equations
Russian Journal of Nonlinear Dynamics, 2006, vol. 2, no. 3, pp. 307-331
Abstract
pdf (4.63 Mb)
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 many-dimensional attractor of the system is described in a region below the synchronization point.
Kuptsov P. V., Kuznetsov S. P., Transition to a synchronous chaos regime in a system of coupled non-autonomous oscillators presented in terms of amplitude equations, Russian Journal of Nonlinear Dynamics, 2006, vol. 2, no. 3, pp. 307-331
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
Abstract
pdf (195.04 Kb)
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
Generalized Dimensions of Feigenbaum's Attractor from Renormalization-Group Functional Equations
Regular and Chaotic Dynamics, 2002, vol. 7, no. 3, pp. 325-330
Abstract
pdf (197.76 Kb)
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
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
Abstract
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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
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, no. 3-4, pp. 90-105
Abstract
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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, no. 3-4, pp. 90-105