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

Grant of Russian Science Foundation No 15-12-20035



The project is carried out in 2015-2017
in the Udmurt State University


Project supervisor:

Sergey Kuznetsov


Participants of the project:

Alexey Borisov

Alexander Kilin

Andrey Tsiganov

Igor Sataev

Ivan Bizyaev

Nadezhda Erdakova

Alexey Kazakov

Vyacheslav Kruglov

Elena Pivovarova

Evgeny Vetchanin

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, Kuzneysov 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. (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. (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

           Kuzneysov 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., Kuzneysov 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., Kuzneysov S.P. Numerical test for hyperbolicity in chaotic systems with multiple time delays. Communications in Nonlinear Science and Numerical Simulation, 2018, 56, 227-239.

           Kuzneysov 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.

           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.

           Tsiganov A.V. Backlund transformations and divisor doubling //Preprint: arXiv:1702.03642 [nlin.SI], 2017, 1-11.