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Vol. 14, no. 3

Vilke V. G.,  Shatina A. V.,  Osipova L.
The classical $N$-body problem in the case when one of the bodies (the Sun) has a much larger mass than the rest of the mutually gravitating bodies is considered. The system of equations in canonical Delaunay variables describing the motion of the system relative to the barycentric coordinate system is derived via the methods of analitical dynamics. The procedure of averaging over the fast angular variables (mean anomalies) leads to the equation describing the evolution of a single Solar system planet’s perihelion as the sum of two terms. The first term corresponds to the gravitational disturbances caused by the rest of the planets, as in the case of a motionless Sun. The second appears because the problem is considered in the barycentric coordinate system and the orbits’ inclinations are taken into account. This term vanishes if all planets are assumed to be moving in one static plane. This term contributes substantially to the Mercury’s and Venus’s perihelion evolutions. For the rest of the planet this term is small compared to the first one. For example, for Mercury the values of the two terms in question were calculated to be 528.67 and 39.64 angular seconds per century, respectively.
Keywords: $N$-body problem, method of averaging, Delaunay variables, orbital elements
Citation: Vilke V. G.,  Shatina A. V.,  Osipova L., The Effect of the Mutual Gravitational Interactions on the Perihelia Displacement of the Orbits of the Solar System’s Planets, Rus. J. Nonlin. Dyn., 2018, Vol. 14, no. 3, pp. 291-300
Kubyshkin E. P.,  Moriakova A. R.
We study equilibrium states and bifurcations of periodic solutions from the equilibrium state of the Ikeda delay-differential equation well known in nonlinear optics. This equation was proposed as a mathematical model of a passive optical resonator in a nonlinear environment. The equation, written in a characteristic time scale, contains a small parameter at the derivative, which makes it singular. It is shown that the behavior of solutions of the equation with initial conditions from the fixed neighborhood of the equilibrium state in the phase space of the equation is described by a countable system of nonlinear ordinary differential equations. This system of equations has a minimal structure and is called the normal form of the equation in the neighborhood of the equilibrium state. The system of equations allows us to pick out one “fast” and a countable number of “slow” variables and apply the averaging method to the system obtained. It is shown that the equilibrium states of a system of averaged equations with “slow” variables correspond to periodic solutions of the same type of stability. The possibility of simultaneous bifurcation of a large number of stable periodic solutions(multistability bifurcation) is shown. With further increase in the bifurcation parameter each of the periodic solutions becomes a chaotic attractor through a series of period-doubling bifurcations. Thus, the behavior of the solutions of the Ikeda equation is characterized by chaotic multistability.
Keywords: Ikeda equation, periodic solutions, bifurcation of multistability, chaotic multistability
Citation: Kubyshkin E. P.,  Moriakova A. R., Features of Bifurcations of Periodic Solutions of the Ikeda Equation, Rus. J. Nonlin. Dyn., 2018, Vol. 14, no. 3, pp. 301-324
Pochinka O. V.,  Loginova A.,  Nozdrinova E.
This article presents a number of models that arise in physics, biology, chemistry, etc., described by a one-dimensional reaction-diffusion equation. The local dynamics of such models for various values of the parameters is described by a rough transformation of the circle. Accordingly, the control of such dynamics reduces to the consideration of a continuous family of maps of the circle. In this connection, the question of the possibility of joining two maps of the circle by an arc without bifurcation points naturally arises. In this paper it is shown that any orientation-preserving source-sink diffeomorphism on a circle is joined by such an arc. Note that such a result is not true for multidimensional spheres.
Keywords: reaction-diffusion equation, source-sink arc
Citation: Pochinka O. V.,  Loginova A.,  Nozdrinova E., One-Dimensional Reaction-Diffusion Equations and Simple Source-Sink Arcs on a Circle, Rus. J. Nonlin. Dyn., 2018, Vol. 14, no. 3, pp. 325-330
Nazarov V. E.,  Kiyashko S. B.
On the basis of the elastic contact model of rough surfaces of solids, a quadraticallybimodular equation of state for micro-inhomogeneous media containing cracks is derived. A study is made of the propagation of elastic single unipolar pulse perturbations and bipolar periodic waves in such media. Exact analytical solutions that describe the evolution of initially triangular pulses and periodic sawtooth waves are obtained. A numerical and graphical analysis of the solutions is also carried out.
Keywords: elastic contact, quadratically-bimodular nonlinearity, pulse perturbation, periodic waves
Citation: Nazarov V. E.,  Kiyashko S. B., Evolution of Pulse and Periodic Elastic Waves in Media with Quadratically-Bimodular Nonlinearity, Rus. J. Nonlin. Dyn., 2018, Vol. 14, no. 3, pp. 331-342
Safronov A.,  Koroteev A. A.,  Filatov N. I.,  Grigoriev A. L.
The influence of long-range interactions on the progress of heat waves in the radiationcooling disperse flow is considered. It is shown that the system exhibits oscillations attendant on the process of establishing an equilibrium temperature profile. The oscillation amplitude and the rate of oscillation damping are determined. The conditions under which the radiation cooling process can be unstable with respect to temperature field perturbations are revealed. The results of theoretical analysis and numerical calculation of the actual droplet flow are compared.
Keywords: disperse flows, radiative heat transfer, long-range interactions, instability
Citation: Safronov A.,  Koroteev A. A.,  Filatov N. I.,  Grigoriev A. L., The Effect of Long-Range Interactions on Development of Thermal Waves in the Radiation-Cooling Dispersed Flow, Rus. J. Nonlin. Dyn., 2018, Vol. 14, no. 3, pp. 343-354
Kevorkov S. S.,  Khamidullin R. K.,  Koroleva (Kikot) I. P.,  Smirnov V. V.,  Gusarova E. B.,  Manevitch L. I.
The results of an experimental and numerical investigation of the dynamics of a string with three uniformly distributed discrete masses are presented. This system can be used as a resonant energy sink for protecting structural elements from the effects of undesirable dynamic loads over a wide frequency range. Preliminary studies of the nonlinear dynamics of the system under consideration showed its high energy capacity. In this paper, we present the results of an experimental study in which a shaker’s table mounted cantilever beam was being protected. As a result, the efficiency of the sink was confirmed, and data were also obtained to refine the mathematical model. It was shown that the experimental data obtained are in good agreement with the results of computer simulation.
Keywords: nonlinear dynamics, nonlinear normal mode, limiting phase trajectory, energy exchange, localization
Citation: Kevorkov S. S.,  Khamidullin R. K.,  Koroleva (Kikot) I. P.,  Smirnov V. V.,  Gusarova E. B.,  Manevitch L. I., Efficiency of a Three-Particle Energy Sink: Experimental Study and Numerical Simulation, Rus. J. Nonlin. Dyn., 2018, Vol. 14, no. 3, pp. 355-366
Morozov A. D.,  Morozov K. E.
We study the role of quasi-periodic perturbations in systems close to two-dimensional Hamiltonian ones. Similarly to the problem of the influence of periodic perturbations on a limit cycle, we consider the problem of the passage of an invariant torus through a resonance zone. The conditions for synchronization of quasi-periodic oscillations are established. We illustrate our results using the Duffing –Van der Pol equation as an example.
Keywords: resonances, quasi-periodic, periodic, synchronization, averaged system, phase curves, equilibrium states
Citation: Morozov A. D.,  Morozov K. E., On Synchronization of Quasiperiodic Oscillations, Rus. J. Nonlin. Dyn., 2018, Vol. 14, no. 3, pp. 367-376
Maslov D. A.,  Merkuryev I. V.
The dynamics of a vibrating ring microgyroscope operating in the forced oscillation mode is investigated. The elastic and viscous anisotropy of the resonator and the nonlinearity of oscillations are taken into consideration. Additional nonlinear terms are suggested for the mathematical model of resonator dynamics. In addition to cubic nonlinearity, nonlinearity of the fifth degree is considered. By using the Krylov – Bogolyubov averaging method, equations containing parameters characterizing damping, elastic and viscous anisotropy, as well as coefficients of oscillation nonlinearity are deduced. The parameter identification problem is reduced to solving an overdetermined system of algebraic equations that are linear in the parameters to be identified. The proposed identification method allows testing at large oscillation amplitudes corresponding to a sufficiently high signal-to-noise ratio. It is shown that taking nonlinearities into account significantly increases the accuracy of parameter identification in the case of large oscillation amplitudes.
Keywords: parameter identification, vibrating ring microgyroscope, nonlinear oscillations
Citation: Maslov D. A.,  Merkuryev I. V., Increase in the Accuracy of the Parameters Identification for a Vibrating Ring Microgyroscope Operating in the Forced Oscillation Mode with Nonlinearity Taken into Account, Rus. J. Nonlin. Dyn., 2018, Vol. 14, no. 3, pp. 377-386
Andreev A. S.,  Peregudova O.
In this paper, the stability and stabilization problems for nonlinear Volterra integrodifferential equations with unlimited delay are considered. The development of the direct Lyapunov method in the study of the limiting properties of the solutions of these equations is carried out by using Lyapunov functionals with a semidefinite time derivative. The topological dynamics of these equations has been constructed revealing the limiting properties of their solutions. The assumption of the existence of a Lyapunov functional with a semidefinite time derivative gives a more complete solution to the positive limit set localization problem. On this basis new theorems on sufficient conditions for the asymptotic stability and instability of the zero solution of nonlinear Volterra integro-differential equations are proved. These theorems are applied to the problem of the equilibrium position stability of the hereditary mechanical systems as well as the regulation problem of the controlled mechanical systems using a proportional-integro-differential controller. As an example, the regulation problem of a mobile robot with three omnidirectional wheels and a displaced mass center is solved using the nonlinear integral controllers without velocity measurements.
Keywords: Volterra integro-differential equation, stability, Lyapunov functional, limiting equation, regulation problem
Citation: Andreev A. S.,  Peregudova O., On the Stability and Stabilization Problems of Volterra Integro-Differential Equations, Rus. J. Nonlin. Dyn., 2018, Vol. 14, no. 3, pp. 387-407
Kiselev O. M.
We obtain criteria for the stability of fast straight-line motion of a wheeled robot using proportional or proportional derivative feedback control. The motion of fast robots with discrete feedback control is defined by the discrete dynamical system. The stability criteria are obtained for the discrete system for proportional and proportional-derivative feedback control.
Keywords: feedback control, stability, robotics
Citation: Kiselev O. M., Stable Feedback Control of a Fast Wheeled Robot, Rus. J. Nonlin. Dyn., 2018, Vol. 14, no. 3, pp. 409-417