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
with prolongation in 2018-2019
Participants of the project:
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.
• 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.
• 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
• 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.
• 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.
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.
• 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.)
• 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.
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