Complex dynamics of nonlinear mechanical
and radio physical systems and its applications

Grant of Russian Science Foundation No 15-12-20035



The project was carried out in 2015-2017 with prolongation in 2018-2019
in the Udmurt State University


Project supervisor:

Sergey Kuznetsov


Participants of the project:

Alexey Borisov

Alexander Kilin

Andrey Tsiganov

Ivan Mamaev

Vyacheslav Kruglov

Elena Pivovarova

Tatjana Ivanova

Elizaveta Artemova

Anton Klekovkin

Former participants:

Igor Sataev

Ivan Bizyaev

Nadezhda Erdakova

Alexey Kazakov

Evgeny Vetchanin

Yury Karavaev

Main Results in 2015

New examples of systems with rough chaos based on coupled oscillators or rotators implementing Anosov dynamics and modification of these systems of self-oscillatory type have been constructed. Mathematical models of the proposed systems are developed including description in terms of geodesic flows on two-dimensional manifolds of negative curvature, and their numerical study has been carried out. Numerical calculations demonstrate that the chaotic dynamics in the introduced self-oscillating systems is associated with hyperbolic attractors, and hence is rough, at least, for a relatively small supercriticality of the self-oscillation mode.

Circuit implementation is proposed for electronic devices corresponding to equations of the two-dimensional problem of the plate motion in a vicious medium at zero buoyancy, constant circulation around the profile, and applied constant external torque, and chaotic dynamics are demonstrated.

For model description of two-dimensional motion of a solid body (plate) in fluid in terms of ordinary differential equations, a methodology is proposed combining the phenomenological approach postulating the general form of equations and the approach developed in the modern nonlinear dynamics for reconstruction of models from the observables obtained from the numerical solution of the two-dimensional problem with the Navier-Stokes equations.

For the controlled motion of an arbitrary two-dimensional body in a fluid assuming a constant circulation around the profile, it is shown that by variations of the position of the inner mass and rotations of the inner rotor the moving body can be directed in a neighborhood of a prescribed spatial point.

Theory of excitation of acoustic waves and oscillations in resonators and periodic structures is developed, where the excited acoustic field is the velocity field, while the sources are represented by vorticity in the flow. For two-dimensional problems the equations of excitation of the acoustic oscillations and waves have been formulated in a form similar to that for electrodynamical resonators and periodic structures in microwave electronics, and for the three-dimensional case the equations are derived corresponding to the electrodynamic theory in the overall structure. On this basis, problems of instability in the interaction of vortices with periodic structures are studied.

A nonholonomic model for a top of special kind in the gravity field is formulated and investigated, which is a generalization of the classical nonholonomic Suslov problem. In the dynamics of the Suslov top conservative chaos has been found, as well as strange attractors, an intermediate type of chaotic behavior (the mixed dynamics). Novel phenomena exhibited by this object are identified and studied, namely, the effect of reversal of rotation and of turnover of the rotating object upside down.

For the problem concerning the motion of a point particle in a potential field in three-dimensional Euclidean space with nonholonomic constraints, particularly for the nonholonomic oscillator and the Heisenberg system, using the Chaplygin reducing multiplier method, a conformal Hamiltonian representation is provided, which reduces the problem to consideration of a particle in potential field on a plane or on a sphere. For the problem with nonholonomic constraint due to Blackall the impossibility of the Hamiltonian reduction is shown.

It is established that a nonholonomic Chaplygin top model demonstrates the scenarios of transition to chaos and destruction of quasi-periodic motions characteristic for the dissipative dynamics, including the period-doubling Feigenbaum bifurcation cascade and transition and torus doubling cascade. In certain parameter areas of the Chaplygin top the possibility of implementing specific "metascenarios" (collections of bifurcation events containing typical scenarios of transition to chaos as their stages) is demonstrated for the evolution of coexisting attractors, including emergence of the "figure-eight" homoclinic attractor and of a specific ring-shaped heteroclinic attractor.

Publications 2015

           Borisov A.V., Mamaev I.S. Symmetries and reduction in nonholonomic mechanics. Regular and Chaotic Dynamics, 2015, 20, №5, 553-604.

           Bizyaev I.A., Borisov A.V., Kazakov A.O. Dynamics of the Suslov problem in a gravitational field: Reversal and strange attractors. Regular and Chaotic Dynamics, 2015, 20, №5, 605-626.

           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, 20, 2015, No 6, 649–666.

           Kuznetsov S.P. On the validity of nonholonomic model of the rattleback. Physics-Uspekhi, 58, 2015, No 12, 1222-1224.

           Kuznetsov A.P., Kuznetsov S.P., Trubetskov D.I. Analogy in interactions of electronic beams and hydrodynamic flows with fields of resonators and periodic structures. Part 1. Izvestiya VUZ. Applied Nonlinear Dynamics, 23, 2015, No 5, 5-40. (Russian.)

           Tsiganov A.V. On Integrable Perturbations of Some Nonholonomic Systems. Symmetry, Integrability and Geometry: Methods and Applications, 11, 2015, 085.

Main Results in 2016

For the first time, a method of computer verification of hyperbolic nature of chaotic attractors based on calculation of angles between expanding, compressing and neutral manifolds for the phase trajectories ("the angle criterion") has been developed for the class of time-delay systems. The hyperbolicity of chaos is substantiated for previously proposed examples of the time-delay systems.

A set of self-oscillatory systems based on interacting rotators with attractors reproducing dynamics of geodesic flow on a surface of negative curvature is proposed. On this base, an electronic circuit operating as a generator of rough chaos is composed of the phase-locked loops as an electronic equivalent to rotators.

Technique of reconstruction of ordinary differential equations on a base of processing time series obtained by numerical solution of the Navier – Stokes equations has been successfully applied for approximate description of the plane problem of motion of a body of elliptic profile in incompressible viscous fluid under action of gravity.

Using approach based on analogy with electrodynamics to interaction of vortex flows with acoustic fields of resonators and periodic structures, the problem of flat vortex tape interacting with a periodic structure of comb type has been analyzed; the dispersion equation of the problem is derived, and hydrodynamic structures similar to microwave electronic devices with crossed fields are proposed.

For a system of two point vortices in hydrodynamic flow excited by external acoustic field the bifurcations are found and analyzed including saddle-node bifurcation, supercritical and subcritical reversible pitch-fork bifurcations, bifurcation of symmetry break, leading to emergence and subsequent transformation of stable regular modes.

For motion of a body in ideal incompressible fluid containing internal movable masses and an internal rotor, controllability is established for different combinations of the control elements. For the case of zero circulation, explicit control actions (gates) are composed to ensure rotations and directed motions. It is proven that the body can be moved out any initial position to any final position using internal motions of two material points on circles of the same radius, or by the inner rotor turning complemented by reciprocating motion of internal masses or circular motions of a single internal mass.

Circuit implementation is developed and a comparative study within framework of numerical calculations and simulating with the Multisim package is provided for parametric oscillator based on a reactive element having exactly the quadratic nonlinear characteristic. The latter allows using the circuit for analog simulation of a two-dimensional problem of motion of an elliptic profile body in a resistant medium in neutral buoyancy.

A novel phenomenon of nonlinear dynamics is described – the strange non-chaotic self-oscillations. An example is provided by a mechanical autonomous system composed of rotating discs with friction transmissions and supplied constant torque that manifests a strange non-chaotic attractor of type, which was discussed so far only for non-autonomous systems with quasi-periodic external driving.

For a dynamically unbalanced ball moving on a horizontal plane with superimposed nonholonomic constraint, scenarios of transition to chaos are discovered associated with destruction of an invariant curve through the Neimark-Sacker bifurcation, and the Feigenbaum period doubling bifurcation scenario. Attractors in the system conserving mechanical energy arise due to existence of domains of compression together with those of expansion of phase volume in the state space of the nonholonomic model.

For paradigmatic nonholonomic system, the Chaplygin sleigh moving on a plane in presence of a weak viscous resistance force under periodic pulses of torque depending on the instant spatial orientation, we demonstrate, discuss and classify regular and chaotic dynamic modes. They include directed average motions and random walks of diffusion type, corresponding, respectively, to regular and chaotic attractors of the map describing dynamics in the 3D space of the rotational angle and generalized velocities.

A new method of constructing Bäcklund transformation for Hamilton-Jacobi equations is proposed, and explicit formulas are obtained for integrable systems on hyperelliptic curves of the first and second kind. The question of applicability of mathematical methods of Hamiltonian dynamics to conformal Hamiltonian vector fields is considered, several examples of such fields arising in the control theory are analyzed; a possibility of introducing a new classification sign for such systems is opened.

Publications 2016

           Kuznetsov S.P., Kruglov V.P. Verification of hyperbolicity for attractors of some mechanical systems with chaotic dynamics. Regular and Chaotic Dynamics, 21, 2016, No 2, 160–174.

           Tsiganov A.V. On a family of Backlund transformations. Doklady Mathematics, 93, 2016, No 3, 292–294.

           Vetchanin E.V., Kazakov A.O. Bifurcations and Chaos in the Dynamics of Two Point Vortices in an Acoustic Wave. International Journal of Bifurcation and Chaos, 26, 2016, No 4, 1650063.

           Grigoryev Yu.A., Sozonov A.P., Tsiganov A.V. Integrability of Nonholonomic Heisenberg Type Systems. Symmetry, Integrability and Geometry: Methods and Applications, 12, 2016, 112.

           Kuptsov P.V., Kuznetsov S.P. Numerical test for hyperbolicity of chaotic dynamics in time-delay systems.. Phys. Rev. E, 94, 2016, No 1, 010201(R). Preprint

           Jalnine A.Yu, Kuznetsov S.P. Strange nonchaotic self-oscillator. Europhysics Letters, 115, No 3, 2016, 30004. Preprint.

           Borisov A.V., Kuznetsov S.P. Regular and Chaotic Motions of a Chaplygin Sleigh under Periodic Pulsed Torque Impacts. Regular and Chaotic Dynamics, 21, 2016, No 7-8, 792–803.

           Borisov A.V., Kazakov A.O., Sataev I.R. Spiral Chaos in the Nonholonomic Model of a Chaplygin Top. Regular and Chaotic Dynamics, 21, 2016, No 7-8, 939–954.

           Borisov A.V., Kazakov A.O., Pivovarova E.N. Regular and Chaotic Dynamics in the Rubber Model of a Chaplygin Top. Regular and Chaotic Dynamics, 21, 2016, No 7-8, 885–901.

           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, 12, 2016, No 2, 223–234. (In Russian.)

           Sataev I.R., Kazakov A.O. Scenarios of transition to chaos in the nonholonomic model of a Chaplygin top. Russian Journal of Nonlinear Dynamics, 12, 2016, No 2, 235–250. (In Russian.)

           Kuznetsov S.P., Borisov A.V., 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, 42, №9, 886–890. (Russian.)

           Bizyaev I.A., Borisov A.V., Mamaev I.S. The Hojman Construction and Hamiltonization of Nonholonomic Systems. Symmetry, Integrability and Geometry: Methods and Applications, 12, 2016, 012.

           Kuznetsov A.P., Kuznetsov S.P. Analogy in interactions of electronic beams and hydrodynamic flows with fields of resonators and periodic structures. Part 2. Self-excitation, amplification and dip conditions. Izvestiya VUZ. Applied Nonlinear Dynamics, 24, 2016, No 2, 5-26. (Russian.)

           Kuznetsov S.P. Lorenz type attractor in electronic parametric generator and its transformation outside the accurate parametric resonance. Izvestiya VUZ. Applied Nonlinear Dynamics, 24, 2016, No 3, 68-87. (Russian.)

           Kuznetsov S.P. From Anosov’s Dynamics on a Surface of Negative Curvature to Electronic Generator of Robust Chaos. Izv. Saratov Univ. (N.S.), Ser. Physics, 16, 2016, No 3, 131–144. (Russian.)

           Vetchanin E.V., Kilin A.A. Controlled motion of a rigid body with internal mechanisms in an ideal incompressible fluid. Proceedings of the Steklov Institute of Mathematics, 2016, 295, 302–332.

           Kuznetsov S.P., Borisov A.V., Mamaev I.S., Tenenev V.A. Reconstruction of model equations to the problem of the body of elliptic cross-section falling in a viscous fluid. VI International Conference GDIS 2016. Book of Abstracts. Moscow – Izhevsk: Institute of Computer Science, 2016. ISBN 978-5-4344-0361-0. P.39-40.

           Sataev I.R., Kazakov A.O. Routes to chaos in the nonholonomic model of Chaplygin top. VI International Conference GDIS 2016. Book of Abstracts. Moscow – Izhevsk: Institute of Computer Science, 2016. P.53-54.

           Grigoryev Yu.A. , Sozonov A.P., Tsiganov A.V. On integrable perturbations of the Brockett nonholonomic integrator. Preprint: arXiv:1603.03528 [nlin.SI], 2016, 1-14.

Main Results in 2017

Methods of computer testing of the hyperbolic nature of attractors have been developed for systems having an arbitrary number of feedback circuits with different delay times, and a substantiation of the hyperbolic nature of chaos has been presented for the first time using these methods for several previously proposed systems with two delays.

A two-dimensional map has been introduced into consideration, which for energy-preserving systems of nonholonomic mechanics may claim the role of a generalized model, similar to the standard Chirikov-Taylor map of conservative Hamiltonian dynamics. The map has been obtained in analytic form for a concrete problem of the Chaplygin sleigh, when the nonholonomic constraint is periodically switched between three sleigh supports. On the phase plane of the map there are a ``chaotic sea’’ and ``islands’’ formed by invariant curves, as in conservative nonlinear dynamics, and attractors and repellers, as in dissipative dynamics, in regions of prevailing compression or stretching of phase volume.

For a Chaplygin sleigh that executes motion on a plane and carries an oscillating internal mass, the possibility of unbounded acceleration under conditions of small oscillations is shown, with the longitudinal velocity of the sleigh being asymptotically proportional to the cubic root of time. In other parameter regions, periodic, quasi-periodic and chaotic motions with a limited variation in the velocity occur; these motions correspond to attractors in the phase space. In the presence of weak friction, acceleration with small oscillations of the internal mass leads to stabilization of the attainable velocity of motion at a fixed level. An alternative mechanism of acceleration in the presence of friction is determined by the effect of parametric excitation of oscillations. Acceleration is unbounded if the line of oscillations of the moving mass passes through the center of mass. If the latter condition is violated, acceleration is bounded, and the steady-state regime is in many cases associated with a chaotic attractor, and the motion of the sleigh turns out to be similar to the process of random walk. The results for the Chaplygin sleigh can be related also to wheeled vehicles, since the imposed nonholonomic constraint is equivalent to the one that is realized by replacing an element of the constraint in the form of a ``skate’’ on the wheel pair with the other supports sliding freely.

A new mechanical system with a hyperbolic attractor has been proposed based on the coupled Froude pendulums excited by applied constant torque and alternately braked by periodic activation of the friction force. If the parameters are selected properly, the attractor of the stroboscopic Poincaré map is a Smale-Williams solenoid characterized by a four-fold increase in the number of turns on each step of the map. This can serve as an example for construction of a new class of systems of different nature with hyperbolic chaos and quasi-periodic dynamics, based on subsystems such that the transmission of oscillating excitation between them occurs in a resonant way due to the fact that the frequencies of small and large oscillations differ by an integer number of times.

A new example of a mechanical system has been introduced into consideration with constraints where hyperchaos characterized by the presence of two positive Lyapunov exponents takes place. The mechanical system is a flat linkage which consists of four cranks and whose free motion is interpreted as a geodesic flow on a compact three-dimensional Riemannian manifold with curvature.

Information has been gathered in the form of arrays of numerical data obtained by numerical solution of the two-dimensional problem of the motion of an elliptic body under the action of gravity force in an incompressible viscous fluid using the Navier-Stokes equations for different ratios of the lengths of principal axes and the coefficients of viscosity. This information has made it possible to obtain ordinary differential equations, corresponding to different motion regimes, using methods based on the idea of reconstructing the finite-dimensional model by processing observable time series by employing the tools of dynamical systems theory. Samples of trajectories have been constructed to reproduce the results of the initial numerical simulation, and charts of regimes of the system of reconstructed equations, where regions of regular and chaotic motions of self-oscillating and of autorotational type have been plotted.

Equations have been formulated for the planar problem of the motion of a body in a viscous fluid in the presence of a given motion of internal masses within the framework of the Kozlov model, where the interaction of the body with environment is taken into account by introducing added masses and viscous friction, which has different coefficients for the longitudinal and transverse motions. Numerical calculations demonstrate the possibility of maintaining on the average unidirectional motion of the body under conditions of zero buoyancy; in this case, the effect persists in the limiting case of large viscosity if the longitudinal and transverse coefficients of friction differ considerably. Also, for a certain choice of parameters one can observe chaotic motions associated with strange attractors and characterized by the presence of a positive Lyapunov exponent.

For the system of equations of motion of two point vortices in a flow with constant uniform vorticity under the action of a given external wave field, the possibility of regular and chaotic regimes corresponding to simple and chaotic attractors has been shown. Bifurcations of fixed points of the Poincaré map that lead to the appearance of different regimes have been investigated and it has been shown that the cascade of period-doubling bifurcations is a characteristic scenario of transition to chaos.

For an asymmetric unbalanced ball (a Chaplygin top), chaotic regimes of rolling in a gravitational field on a plane without slipping have been found and investigated. These regimes correspond to homoclinic strange attractors of discrete spiral type (discrete attractors of Shilnikov type) for a corresponding three-dimensional Poincaré map that in the general case has no smooth invariant measure. Also, chaotic regimes and attractors of different types have been found for the Chaplygin top with a nonholonomic constraint that ensures the absence of spinning and slipping at the point of contact. A comparative analysis has been made of the dynamical properties of both models. It is shown that the dynamics of the system in absolute space and the behavior of the point of contact in the presence of strange attractors depends considerably on the characteristics of the attractor and can be either chaotic or close to quasi-periodic behavior.

For a top in the form of a truncated ball under the assumption of absence of sliding and rotation of the body about the vertical at the point of contact (the ``rubber’’ model of a nonholonomic constraint) all motions can be divided into three types: rolling within the framework of the disk model, rolling within the framework of the ball model, and rolling with periodic transition between these two models. Although no transitions occur between these three types during motion within the framework of the ``rubber’’ model, they become possible in the case where friction forces are introduced. In this case, the system exhibits both dynamical effects and a retrograde turn of the disk or a turnover of the top.

Bearing in mind the development of quantitative characteristics of the translational motion of mobile systems in situations where the dynamics of reduced equations (for generalized velocities) is regular or chaotic, a number of model problems of the motion of the Chaplygin sleigh have been considered. For a quantitative characteristic of the translational motion in situations of chaotic and regular dynamics in the space of generalized velocities, the following quantities have been introduced: the average velocity, average angular velocity, diffusion coefficient, and the coefficient of diffusion by the angle, which have been found numerically for model problems depending on the parameter of intensity of external periodic driving. For the situation where chaotic dynamics leads to isotropic random motions such as two-dimensional random walk in a laboratory coordinate system, an asymptotic Rayleigh distribution for the distance travelled and uniform distribution for azimuth angles take place.

It has been proved that cryptographic protocols based on the arithmetic of divisors are canonical transformations of different valencies which preserve the form of the Hamilton –Jacobi equations, i.e., Bäcklund self-transformations. It is shown that the ``cryptograms’’, which are new canonical variables on phase space, can be efficiently used to construct integrable generalizations of well-known systems and to construct new integrable systems with integrals of motion of higher degrees within the framework of the Jacobi method. Examples of new Hamiltonian integrable systems with integrals of motion of the sixth, fourth and third degree in momenta on a plane, a sphere and an ellipsoid have been constructed.

Publications 2017

           Kuznetsov S.P. Regular and chaotic motions of the Chaplygin sleigh with periodically switched location of nonholonomic constraint. Europhysics Letters, 118, No 1, 2017, 10007.

           Bizyaev I.A., Borisov A.V., Kuznetsov S.P. Chaplygin sleigh with periodically oscillating internal mass. Europhysics Letters, 119, No 6, 2017, 60008.

           Jalnine A.Yu., Kuznetsov S.P. Autonomous Strange Nonchaotic Oscillations in a System of Mechanical Rotators. Regular and Chaotic Dynamics, 22, 2017, No 3, 210–225.

           Kilin A.A., Pivovarova E.N. The Rolling Motion of a Truncated Ball Without Slipping and Spinning on a Plane. Regular and Chaotic Dynamics, 22, 2017, No 3, 298-317.

           Kuptsov P.V., Kuznetsov S.P. Numerical test for hyperbolicity in chaotic systems with multiple time delays. Communications in Nonlinear Science and Numerical Simulation, 2018, 56, 227-239. (Preprint.)

           Kuznetsov S.P. Chaos in three coupled rotators: From Anosov dynamics to hyperbolic attractors. Indian Academy of Sciences Conference Series, 2017, 1, No 1, 117-132.

           Kuznetsov S.P. Regular and Chaotic Dynamics of a Chaplygin Sleigh due to Periodic Switch of the Nonholonomic Constraint.
Regular and Chaotic Dynamics, 2018, 23, No 2, 178–192.

           Gonchenko A.S., Gonchenko S.V., Kazakov A.O., Кozlov A.D. Mathematical theory of dynamical chaos and its applications: Review. Part 1. Pseudohyperbolic attractors. Izvestiya VUZ. Applied Nonlinear Dynamics, 25, 2017, No 2, 4-36. (In Russian.)

           Jalnine A. Y., Kuznetsov S. P. Autonomous strange non-chaotic oscillations in a system of mechanical rotators. Rus. J. Nonlin. Dyn., 2017, 13, No 2, 257-275. (In Russian.)

           Borisov A. V., Kazakov A. O., Pivovarova E. N. Regular and chaotic dynamics in the rubber model of a Chaplygin top Rus. J. Nonlin. Dyn., 2017, 13, No 2, 277-297. (In Russian.)

           Kuznetsov S. P., Kruglov V. P. On some simple examples of mechanical systems with hyperbolic chaos. Proceedings of the Steklov Institute of Mathematics, 297, 208-234.

           Kuznetsov S.P. Complex dynamics of Chaplygin sleigh due to periodic switch of the nonholonomic constraint location. The International Scientific Workshop "Recent Advances in Hamiltonian and Nonholonomic Dynamics" (Moscow, Dolgoprudny, Russia, 15-18 June 2017). Book of Abstracts. Moscow – Izhevsk: Institute of Computer Science, 2017. ISBN 978-5-4344-0445-7. P.53-56.

           Kuznetsov S.P. Design principles and illustrations of hyperbolic chaos in mechanical and electronic systems. Proceedingsof the International Symposium "Topical Problems of Nonlinear Wave Physics" (Moscow – St. Petersburg, Russia, 22 – 28 July, 2017). Institute of Applied Physics of RAS, Nizhny Novgorod, 2017. P.44.

Main Results in 2018

A model of controlled spherical robot with an axisymmetric pendulum drive equipped with a feedback system that suppresses uncompensated oscillations of the pendulum at the final stage of motion, has been developed. According to the proposed approach, the feedback depends on the phase variables and does not depend on the trajectory type. The results of experimental studies confirm possibility of using the proposed controllers to stabilize the movements, and demonstrate their effectiveness.

Dynamics of the rolling dynamically asymmetric unbalanced ball (Chaplygin top) on a horizontal plane under the action of a periodic gyrostatic moment has been studied in the framework of the rubber body model, that is, under the condition that there is no slip and spinning at the point of contact. It is shown that for certain values of the system parameters and the character of the time dependence of the gyrostatic moment, there acceleration of the system takes place, that is, an unlimited increase in the kinetic energy. Dependence of the acceleration on the system parameters and the initial conditions is analyzed. On the basis of studies of dynamics of the frozen system, a hypothesis has been suggested concerning general mechanism of acceleration due to periodic driving in nonholonomic systems.

A mathematical model for a nonholonomic mechanical multicomponent system has been formulated that is a platform sliding along a two-dimensional surface, so that a fixed point of the platform is prohibited to move across a specific direction, and with masses attached performing a given movement relative to the platform; it corresponds to a generalization of the Chaplygin sleigh model. For motions in the presence of weak longitudinal friction due to small oscillations of a single internal mass, it has been shown that the acceleration of the motion, which is straight in average, the speed of the sleigh stabilizes at a certain level. In the case of parametric excitations of oscillations, when the oscillating mass is comparable to the mass of the main platform, the increase in the kinetic energy of the sleigh appears to be limited if the oscillation line of the moving mass is displaced from the center of mass. The sustained mode in this case may correspond to a chaotic attractor, and the movement of the sleigh in the laboratory frame has a character of a two-dimensional random walk. The presence of chaotic attractors for mobile systems with moving internal masses makes it possible to apply chaos control methods to the diffusion-type motions, which, due to the sensitivity of chaos to small perturbations, can be carried out with arbitrarily small targeted effects.

Comparison of the dynamics for the nonholonomic model of the Chaplygin sleigh and for its modification with longitudinal viscous friction shows that this radically changes the character of movements in parameter domains of quasi-conservative behavior, instead of which multistability is realized that is coexistence of many attractive cycles in the state space. On the other hand, in the parameter domain where fractal attractors occur, the weak dissipation gives rise only to quantitative changes in the characteristics of chaos, without significant change in the structure of attractors.

As a new approach to the problem of creating mobile devices that move in a volume or on a surface of fluid medium due to motions of internal masses, the idea is proposed to depart from the nonholonomic model of the Chaplygin sleigh, replacing the nonholonomic constraint at the point of its application by strong viscous friction in the direction transverse to the direction of allowed sliding. On the basis of numerical simulation, the main types of dynamic behavior are revealed, and it is shown that in the parameter space of the nonholonomic model and of the system with viscous friction, the arrangement of regions of regular and chaotic dynamics is similar, as well as the bifurcation scenarios leading to chaos. It is shown that the effect of accelerating the platform motion of due to small oscillations of the internal mass persists in the region of low velocities, but the velocity growth tends to saturate. It is shown that when a nonholonomic constraint is replaced by the viscous friction, fractal chaotic attractors continue to exist with minor changes, while the “fat attractors” corresponding to quasi-conservative dynamics are destroyed with formation of many coexisting regular attractors in the form of attracting cycles. It is shown that motions of the platform corresponding to a two-dimensional random walk, caused by the strange attractors, persist in the system with friction, but are characterized by smaller diffusion coefficients than those for the nonholonomic model. The effect of parametric resonance, which in a nonholonomic model can lead to unlimited growth of the kinetic energy of the platform, in a system with viscous friction is characterized by saturation of the parametric instability, so that the sustained average velocity of the platform relative to the medium appears to be limited.

Model systems are studied composed of rotators with constraint given by the condition of zero sum of the cosines of the rotation angles. The system of three rotators, which corresponds to geodesic flow on the two-dimensional surface of Schwartz, demonstrates chaos characterized by one positive Lyapunov exponent, and systems of four and five rotators that are associated with geodetic flows on three-dimensional and four-dimensional manifolds with curvature have, respectively, two and three positive exponents ("hyperchaos"). An algorithm has been implemented that allows calculating the sectional curvature of the manifold in the course of numerical simulating the dynamics at points of the trajectory. In contrast to the case of three rotators, where the curvature of the manifold is negative (except for a finite number of points) and the Anosov flow is realized, i.e. the hyperbolic type of dynamics, in the case of four and five rotators, the condition of negative sectional curvature is not fulfilled, geodesic flows cannot be classified as Anosov systems, and the dynamics is non-hyperbolic.

For models composed of coupled self-oscillating elements, respective amplitude equations, and equations linearized in respect to deviation of the amplitudes from the limiting cycle, dynamic phenomena correlate with those in the Topaj-Pikovsky model of coupled phase oscillators, just as it is in the case of systems of nonholonomic mechanics modified to their analogues when taking into account viscous friction. The correspondence of dynamic behavior with the Topaj-Pikovsky model can be traced at finite observation times, but in the sense of the asymptotic regimes it can be very different. In particular, quasi-conservative dynamics are destroyed, and instead of the “chaotic sea” and regular cyclic motions, multistability arises that is coexistence of a set of attracting cycles.

An approach is developed that allows to verify the presence or absence of pseudo-hyperbolic dynamics of Shilnikov and Turaev by numerical calculations for specific systems, which consists in checking for trajectories on the attractor that there are no tangencies between the subspaces of the perturbation vectors, one of which expands the phase volume, and the second compresses. New quantifiers have been introduced, which are modifications of Lyapunov exponents together with the Lyapunov vectors associated with them. The technique was tested on classical attractors of Lorenz and Rössler, the first of which does belong to the category of pseudohyperbolic, and the second does not. The pseudo-hyperbolic nature of the attractors, realized with a certain choice of parameters, was confirmed for the three-dimensional Hénon map and for the system of ordinary differential equations, which is a four-dimensional generalization of the Lorenz model.

For the first time, a physically implementable example of a system with delay where a pseudohyperbolic attractor occurs is presented, which is confirmed at the level of numerical calculations using the method of analyzing the intersection angles for subspaces of the perturbation vectors that expand and compress the phase volume. Using the tools of nonlinear dynamics, including waveform plots, phase portraits of attractors in two-dimensional projections, analysis of spectra and Lyapunov exponents, a close analogy of the attractor of the proposed system with the classical Lorenz attractor has been demonstrated.

For a model of mechanical oscillatory system in the form of coupled Froude pendulums, excited due to the applied torque in the case of the friction force decrease depending on the angular velocity being alternately slowed down by periodic application of brake shoed, the possibility of implementing various dynamic modes accompanied by oscillations or rotations of the pendulums is shown. Among them are periodic modes represented by attracting limit cycles, quasi-periodic modes corresponding to attracting invariant tori, strange attractors with one and two positive Lyapunov exponents (chaos and hyperhaos). The hyperbolicity of chaotic attractors was tested by analyzing the distribution of the intersection angles of stable and unstable manifolds by processing numerical results for typical trajectories, and it was shown that depending on the parameters the chaotic attractors of the system can be either non-hyperbolic (tangencies of the manifolds occur) or hyperbolic (no tangencies). Hyperbolic chaos takes place situation when the transmission of the oscillatory excitation between the subsystems is resonant due to the two-fold difference in the frequencies of small and large-amplitude oscillations. In this mode, the attractor in the Poincaré section is a Smale – Williams solenoid, and it is characterized by structural stability, that is, it persists with small variations of parameters.

As an example of constructing n-point finite difference equations for integrable systems, the Euler top discretization is considered. It is shown how the divisors of the intersection of elliptic and hyperelliptic curves with straight lines, quadrics and cubes generate families of integrable discrete mappings. The Bäcklund transformations for the Lagrange top and the Hénon-Heiles system are considered. It is proved that the multiplication of the divisors of curves by the scalar used in modern cryptography generates transformations that preserve the forms of the integrals of motion and the Poisson brackets up to a scalar factor. Using the Korkin-Goryachev-Bobylev-Steklov method, several classes of systems with non-potential forces with periodic trajectories were studied, which allow constructing four families of superintegrable systems on a plane that permit separation of variables in the Cartesian coordinate system.

Publications 2018

           Bizyaev I.A., Borisov A.V., Kuznetsov S.P. The Chaplygin sleigh with friction moving due to periodic oscillations of an internal mass. Nonlinear Dynamics, 2019, 95, iss.1, 699–714. (Preprint.)

           Kruglov V.P., Kuznetsov S.P. Hyperbolic chaos in a system of two Froude pendulums with alternating periodic braking. Communications in Nonlinear Science and Numerical Simulation, 2019, 67, 152-161. (Preprint.)

           Tsiganov A.V. Backlund transformations and divisor doubling. Journal of Geometry and Physics, 2018, 126, 148-158. (Preprint.)

           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, 23, 2018, Nos 7-8, 908–932.

           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, 23, 2018, Nos 7-8, 803–820.

           Ivanova T.B., Kilin A.A., Pivovarova E.N. Controlled Motion of a Spherical Robot with Feedback. I. Journal of Dynamical and Control Systems, 2018, 24, 497–510.

           Tsiganov A.V. On Discretization of the Euler Top. Regular and Chaotic Dynamics, 23, 2018, No 6, pp.785-796.

           Kilin A.A., Pivovarova E.N. Chaplygin Top with a Periodic Gyrostatic Moment. Russian Journal of Mathematical Physics, 25, 2018, No 4, pp.517–532.

           Kuznetsov S.P., Kuptsov P.V. Lorenz Attractor in a System with Delay: an Example of Pseudogyperbolic Chaos. Izvestiya of Saratov University. New series. Series Physics, 16, 2018, iss.3, 162-176. (In Russian.)

           Kuznetsov S.P. Chaos and hyperchaos of geodesic flows on curved manifolds corresponding to mechanically coupled rotators: Examples and numerical study. Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp’yuternye Nauki, 28, 2018, iss.4, 565–581. (In Russian.) English translation.

           Kruglov V.P., Kuznetsov S.P. Smale-Williams attractor in a system of alternately oscillating coupled Froude pendulums. VII International Conference GDIS 2018. Book of Abstracts. Moscow – Izhevsk: Institute of Computer Science, 2018. ISBN 978-5-4344-0520-1. P.56-58.

           Kuznetsov S.P., Bizyaev I.A., Borisov A.V. Self-acceleration of Chaplygin sleigh. VII International Conference GDIS 2018. Book of Abstracts. Moscow – Izhevsk: Institute of Computer Science, 2018. P.65-67.

           Kuznetsov S.P. Chaplygin sleigh. Proceedings of the XVII Winter School-Workshop on Radiophysics and Microwave Electronics, 5–10 Feb. 2018, Saratov, publishing house “Nauka”. P.34-35. (In Russian.)

           Kruglov V.P., Kuznetsov S.P. The Smale - Williams attractor in a system of coupled Froude pendulums with alternating braking. Nanoelectronics, nanophotonics and nonlinear physics. Abstracts of the XIII All-Russian Conference of Young Scientists. Saratov, publishing house "Techno-Decor", 2018. ISBN 978-5-6041624-1-5. P.141-142. (In Russian.)

           Doroshenko V.M., Kruglov V.P., Kuznetsov S.P. Hyperbolic chaos in coupled self-oscillatory systems, functioning with the excitation of relaxation self-oscillations. Wave phenomena in inhomogeneous media. Collection of works of the XVI All-Russian School-Seminar named after A.P. Sukhorukov. Nonlinear dynamics and information systems. Krasnovidovo, Moscow region, 2018, p.13-16.

Main Results in 2019

We studied rolling of a homogeneous heavy ball without slipping on a surface of rotating cylinder and on a surface of a cone rotating around its axis of symmetry, in a nonholonomic formulation without dissipation and in a presence of rolling friction torque proportional to the angular velocity of the ball. It is shown that in the case of non-dissipative rolling without slipping at the contact point, the resulting systems of five differential equations are integrable in both cases, and on the set of levels of the first integrals they reduce to quadratures. A bifurcation analysis of both systems is carried out. When rolling on a cone, conditions were found under which the trajectory of the center of mass can lie in one or two limited zones (bands). When rolling on the cylinder with friction, all trajectories shift downward over time, and the ball falls.

A mathematical model of motion control of a dynamically asymmetric unbalanced ball (Chaplygin top) on the plane has been formulated and studied, where there is unbounded acceleration of the motion of this mobile nonholonomic system, achieved by a periodic variation of the gyrostatic moment. A general hypothesis is formulated on the mechanism of acceleration on a plane of spherical bodies due to periodic variations of the system parameters. The fundamental distinction between acceleration in nonholonomic systems and Fermi acceleration in Hamiltonian systems is outlined.

A numerical study of Chaplygin sleigh motion on a plane in a potential well was carried out, namely in a one-dimensional well (channel) and in a potential with rotational symmetry. At relatively small initial energies the resulting motions turn out to be quasiperiodic, and at sufficiently high energies chaotic motions become typical. At high energies a behavior is realized similar to that observed in conservative dynamics (islands of regularity and a chaotic sea), and at moderate energies, it is similar to that observed in dissipative systems (regular and strange attractors at the constant energy level). This is a new, simpler example of a nonholonomic system, where phenomena characteristic of the Celtic stone or Chaplygin top appear, reflecting the specific nature of nonholonomic systems, which occupy an intermediate position between conservative and dissipative dynamics.

A model of nonholonomic system with parametric excitation was investigated at the level of mathematical proofs that is a Chaplygin sleigh in the case when control is achieved by using a rotor with alternating angular momentum. The existence of trajectories is proved for which the translational speed of the sleigh increases indefinitely with time asymptotically as the power of 1/3. It is shown that taking into account viscous friction with non-degenerate Rayleigh function, unbounded acceleration does not occur, and trajectories of the system asymptotically converge to a limit cycle.

A mathematical model is formulated for motion of an elliptical body in a fluid under periodic external force and torque, in a reference frame associated with the body. The dynamics is studied in cases of an ideal and viscous fluid. In the case of ideal fluid the equations have a standard invariant measure, and if the profile of the body is circular, the equations are integrable explicitly. In the case when there is no external torque, the profile is elliptical, and the force acts along one of the main axes of the body, the equations allow a first integral. At a fixed level of this integral a system can manifest chaotic behavior. In the case of viscous fluid it is shown that the translational and angular velocities are limited functions of time due to dissipation. Limit cycles (fixed points of a return map), attracting tori (invariant curves of the map), and strange attractors of the Feigenbaum scenario can occur in the system.

A Hamiltonian model of a lattice of locally coupled oscillators was considered, which has invariant manifolds with asymptotic dynamics exactly equivalent to the Topaj–Pikovsky model. The Hamiltonian model describes spatial modes of the nonlinear Schrödinger equation with periodic tilted potential. Dynamics manifests all the distinguishing properties of the Topaj–Pikovsky phase lattice, related to time reversibility. The stability of trajectories belonging to invariant manifolds is investigated. A system of coupled self-oscillating elements is studied, from which the Topaj–Pikovsky model is obtained in the approximation that takes into account only variations of phases of the oscillators. It turns out that the phenomenology of the Topaj–Pikovsky model and of the more accurate models are correlated in the same way as is the case of nonholonomic mechanics systems and their analogues obtained by taking friction into account.

A mathematical model is formulated describing the Froude pendulum that is capable of self-oscillations due to its placement on a rotating shaft, with the periodic braking by the switching on and off the friction force, and with weak delayed feedback. The regions of chaotic dynamics in the parameter space were revealed using Lyapunov exponents and analysis of topological properties of the mapping of the phase, which made it possible to visualize the region of a hyperbolic attractor of the Smale–Williams type with good accuracy. On the plane the coefficient of delayed feedback – an activity parameter at the buildup stage of the oscillations of the pendulum, one can clearly see the areas where the delayed feedback provides control and anti-control, up to obtaining the hyperbolic chaos.

A possibility is shown of obtaining chaos due to a robust hyperbolic attractor in the ring chain of pendulums with dissipation, if the frequency of the vertical oscillations of the suspension periodically switches, providing alternating parametric excitation of one or the other oscillatory mode. By varying the number of elements in the chain and the mass distribution of the pendulums, one can obtain systems with various integer expansion factors along the angular coordinate of Smale–Williams solenoids.

A non-Hamiltonian vector field is studied that arises when considering rolling of Chaplygin ball on a horizontal plane that rotates with a constant angular velocity. In two special cases the field is expressed in terms of Hamiltonian vector fields using non-algebraic deformation of the canonical Poisson bivector on the algebra of the group of motions of three-dimensional Euclidean space. Unlike systems of Chaplygin, Veselova, Routh and others, this deformation of the Poisson brackets is not trivial, as there is no rational variable change leading to a standard Lie–Poisson bivector on the original algebra e(3). This result allows expanding the list of possible deformations of the Turiel type arising in mathematical description of real systems. In a particular case of a dynamically symmetric ball the separation variables, compatible Poisson brackets, the algebra of Haantjes operators, and the Lax matrix are calculated. The separation variables obtained in this way can also be found directly by using ball symmetry, similar to the Lagrange approach of a symmetric heavy top.

A brief review is presented on dynamics of a point moving on a paraboloid, both under the action of gravity and without it; the paraboloid can rotate around a vertical axis with a constant angular velocity, in the cases of dry (Coulomb) and viscous friction. A review is presented of the problem of rolling spherical shell with a rotating frame inside, on which rotors are mounted, and the center of mass of the entire system is located at the geometric center of the shell. The model of “rubber” nonholonomic constraint (prohibiting spinning and slipping near the contact point) and the classical nonholonomic model (where only slipping is prohibited) are discussed. A review is presented discussing systems in the form of circular lattices of elements of various types manifesting Smale–Williams attractors in the Poincaré section. All examples are united by a common approach to obtaining the hyperbolic dynamics, which consists in arranging the situations of such kind that spatial phases for the resulting patterns ln a characteristic time period is transformed in accordance with expanding mappings.

Publications 2019 г.

           Borisov A.V., Ivanova T.B., Kilin A.A., Mamaev I.S. Nonholonomic rolling of a ball on the surface of a rotating cone. Nonlinear Dynamics, 2019, 97, No 2, pp.1635-1648.

           Kuznetsov S.P. Chaotic dynamics of pendulum ring chain with vibrating suspension . Izvestiya VUZ. Applied Nonlinear Dynamics, 2019, 27, No 4, pp.99-113. (Russian.)

           Kuznetsov S.P., Sedova Yu.V. Robust hyperbolic chaos in Froude pendulum with delayed feedback and periodic braking. International Journal of Bifurcation and Chaos, 29, 2019, iss. 12, (Preprint.)

           Borisov A.V., Kilin A.A., Pivovarova E.N. Speedup of the Chaplygin Top by Means of Rotors. Doklady Physics, 64, 2019, No 3, 120-124.

           Tsiganov A.V. Hamiltonization and Separation of Variables for a Chaplygin Ball on a Rotating Plane. Regular and Chaotic Dynamics, 24, 2019, No 2, 171–186.

           Borisov A.V., Kilin A.A., Mamaev I.S. A Parabolic Chaplygin Pendulum and a Paul Trap: Nonintegrability, Stability, and Boundedness. Regular and Chaotic Dynamics, 24, 2019, No 3, 329–352.

           Bizyaev I.A., Borisov A.V., Mamaev I.S. Different Models of Rolling for a Robot Ball on a Plane as a Generalization of the Chaplygin Ball Problem. Regular and Chaotic Dynamics, 24, 2019, No 5, 560-582.

           Bizyaev I.A., Borisov A.V., Kozlov V.V., Mamaev I.S. Fermi-like acceleration and power-law energy growth in nonholonomic systems. Nonlinearity, 32, 2019, No 9, 3209–3233.

           Kuznetsov S.P. Complex Dynamics in Generalizations of the Chaplygin Sleigh. Russian Journal of Nonlinear Dynamics, 2019, 15, No. 4, 551–559.

           Kuznetsov S.P., Kruglov V.P. Dynamics of Phases and Chaos in Models of Locally Coupled Conservative or Dissipative Oscillators. International Conference " Scientific Heritage of S.A. Chaplygin. Nonholonomic Mechanics, Vortex Structures and Hydrodynamics" (Cheboksary, Russia, 2–6 June 2019). Preprint nlin. arXiv: 1906.10451.

           Kuznetsov S.P. Some Lattice Models with Hyperbolic Chaotic Attractors. International Conference " Scientific Heritage of S.A. Chaplygin. Nonholonomic Mechanics, Vortex Structures and Hydrodynamics" (Cheboksary, Russia, 2–6 June 2019). Preprint nlin. arXiv: 1909.01896.

           Kuznetsov S.P., Kruglov V.P., Sedova Yu.V. Mechanical Systems with Hyperbolic Chaotic Attractors Based on Froude Pendulums. International Conference " Scientific Heritage of S.A. Chaplygin. Nonholonomic Mechanics, Vortex Structures and Hydrodynamics" (Cheboksary, Russia, 2–6 June 2019). Preprint nlin. arXiv: 1909.01155.

           Kruglov V.P., Kuznetsov S.P. Topaj - Pikovsky involution and asymptotic trajectories in the lattice of locally coupled conservative oscillators. Nanoelectronics, nanophotonics and nonlinear physics. Abstracts of XIV All-Russian Conference of young scientists. “Techno-Decor”, Saratov, 2019, p.125-126. (In Russian.)

           Kuznetsov S.P. Hyperbolic chaos in lattice models. Nanoelectronics, nanophotonics and nonlinear physics. Abstracts of XIV All-Russian Conference of young scientists. “Techno-Decor”, Saratov, 2019, p.131-132. (In Russian.)

           Kuznetsov S.P., Sedova Yu.V. Hyperbolic chaos in a self-oscillating system with sine non-linearity and delayed feedback. Nanoelectronics, nanophotonics and nonlinear physics. Abstracts of XIV All-Russian Conference of young scientists. “Techno-Decor”, Saratov, 2019, p.229-230. (In Russian.)

           Kruglov V.P., Kuznetsov S.P. A chain of locally connected conservative oscillators with Topaj - Pikovsky involution. Materials of the XII International School-Conference "Chaotic self-oscillation and formation of structures" (ХАОС-2019). Saratov: OOO "Publishing Center" Nauka", p.59. (In Russian.)

           Sedova Yu.V., Kruglov V.P., Kuznetsov S.P. Rough hyperbolic chaos in systems based on the Froude pendulum. Materials of the XII International School-Conference "Chaotic self-oscillation and formation of structures" (ХАОС-2019). Saratov: OOO "Publishing Center" Nauka", p.95. (In Russian.)