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Vol. 14, no. 2

Rozental R. M.,  Isaeva O. B.,  Ginzburg N. S.,  Zotova I. V.,  Sergeev A. S.,  Rozhnev A. G.
Within the framework of the nonstationary model with nonfixed field structure, we investigate the model of a 3-mm band gyroklystron with delayed feedback. It is shown that both chaotic and hyperchaotic generation regimes are possible in this system. The chaotic regime due to a Feigenbaum period-doubling cascade is characterized by one positive Lyapunov exponent. Further transition to hyperchaos is determined by the appearance of another positive exponent in the Lyapunov spectrum. The correlation dimension of the corresponding attractors reaches values of about 3.5.
Keywords: chaos, hyperchaos, Lyapunov exponents, gyroklystron
Citation: Rozental R. M.,  Isaeva O. B.,  Ginzburg N. S.,  Zotova I. V.,  Sergeev A. S.,  Rozhnev A. G., Characteristics of Chaotic Regimes in a Space-distributed Gyroklystron Model with Delayed Feedback, Rus. J. Nonlin. Dyn., 2018, Vol. 14, no. 2, pp. 155-168
Morozov K. E.
A nonautonomous analogue of the FitzHugh–Nagumo model is considered. It is supposed that the system is transitory, i.e., it is autonomous except on some compact interval of time. We first study the past and future vector fields that determine the system outside the interval of time dependence. Then we build the transition map numerically and discuss the influence of the transitory shift on the solutions behavior.
Keywords: FitzHugh – Nagumo model, transitory system, separatrix, limit cycles, attractors
Citation: Morozov K. E., Transitory Shift in the FitzHugh – Nagumo Model, Rus. J. Nonlin. Dyn., 2018, Vol. 14, no. 2, pp. 169-177
Smirnov V. V.,  Kovaleva M. A.,  Manevitch L. I.
We present an analysis of torsion oscillations in quasi-one-dimensional lattices with periodic potentials of the nearest neighbor interaction. A one-dimensional chain of point dipoles (spins) under an external field and without the latter is the simplest realization of such a system. We obtained dispersion relations for the nonlinear normal modes for a wide range of oscillation amplitudes and wave numbers. The features of the short wavelength part of the spectrum at large-amplitude oscillations are discussed. The problem of localized excitations near the edges of the spectrum is studied by the asymptotic method. We show that the localized oscillations (breathers) appear near the long wavelength edge, while the short wavelength edge of the spectrum contains only dark solitons. The continuum limit of the dynamic equations leads to a generalization of the nonlinear Schrödinger equation and can be considered as a complex representation of the sine-Gordon equation.
Keywords: essentially nonlinear systems, coupled pendulums, nonlinear normal modes, limiting phase trajectories
Citation: Smirnov V. V.,  Kovaleva M. A.,  Manevitch L. I., Nonlinear Dynamics of Torsion Lattices, Rus. J. Nonlin. Dyn., 2018, Vol. 14, no. 2, pp. 179-193
Sergeev K. S.,  Dmitriev S. V.,  Korznikova E. A.,  Chetverikov A. P.
The dynamics of a triangular lattice consisting of active particles is studied. Particles with nonlinear friction interact via nonlinear forces of Morse potential. Nonlinear friction slows down fast particles and accelerates slow ones. Each particle interacts mainly with the nearest neighbors due to the choice of the cut-off radius.
Stationary modes (attractors) and metastable states of the lattice are studied by methods of numerical simulation.
It is shown that the main attractor of the system under consideration is the so-called translational mode — the state with equal and unidirectional velocities of all particles. For some parameter values translational modes with defects in the form of vacancies and interstitial particles are possible.
Metastable localized states are presented by the plane soliton-like waves (M-solitons) with inherent velocity and density maxima. The lifetime of such states depends on the lattice parameters and the wavefront width. All metastable states transform into the translational mode after a transient process.
Keywords: lattices, active particles, solitons, Morse potential
Citation: Sergeev K. S.,  Dmitriev S. V.,  Korznikova E. A.,  Chetverikov A. P., Stationary Modes and Localized Metastable States in a Triangular Lattice of Active Particles, Rus. J. Nonlin. Dyn., 2018, Vol. 14, no. 2, pp. 195-207
Usanov D. A.,  Rytik A. P.,  Gagarina A. V.,  Kutikova O. Y.
This paper explores the phenomenon of synchronization of concentration self-oscillations of the Briggs – Rauscher reaction under external periodic exposure to white light. A high sensitivity of the self-oscillating reaction regime to periodic light irradiation is shown. The paper presents the dependence of the synchronization band on light power, establishes the limiting values of external light irradiation power at which self-oscillations “go off”. The 0.04–0.1 Hz synchronization range of concentration oscillations under external light irradiation is determined. The phenomenon of frequency pulling of concentration oscillations occurred under external light irradiation with frequencies ranging from 0.029 to 0.039 Hz before synchronization and from 0.1 to 0.14 Hz after synchronization. It was established that, as the power of radiation from an external source was increased, the range of synchronization shifted to the low-frequency region of concentration oscillations, while the width of the range changed only slightly. When the luminous flux from the illumination source was less than 250 lm, no synchronization of the self-oscillating reaction regime arose, and when the luminous flux from the source was more than 5000 lm, the self-oscillatory process of the Briggs – Rauscher reaction ceased.
Keywords: self-oscillating reaction, synchronization, phenomenon of frequency pulling, threshold effect, oscillation phase, cessation of the self-oscillating regime
Citation: Usanov D. A.,  Rytik A. P.,  Gagarina A. V.,  Kutikova O. Y., The Phenomenon of Frequency Pulling, Synchronization and Cessation of Self-oscillations of the Briggs – Rauscher Reaction Under Periodic Light Irradiation, Rus. J. Nonlin. Dyn., 2018, Vol. 14, no. 2, pp. 209-216
Kalyakin L. A.
This paper is concerned with a system of three nonlinear differential equations, which is a mathematical model for a system of nuclear spins in an antiferromagnet. The model has arisen in recent physical studies and differs from the well-known and well-understood Landau – Lifshitz and Bloch models in the manner of incorporating dissipation effects. It is established that the system under consideration is related to the Landau – Lifshitz system by the passage to the limit only on one invariant sphere. The initial equations contain three dimensionless parameters. Equilibrium points and their stability are examined depending on these parameters. The position of the bifurcation surface is found in the parameter space. It is proved that the corresponding equilibrium is of saddle-node type. Exact statements are illustrated by results of numerical experiments.
Keywords: nonlinear equations, equilibrium, stability, bifurcation
Citation: Kalyakin L. A., Analysis of a Mathematical Model for Nuclear Spins in an Antiferromagnet, Rus. J. Nonlin. Dyn., 2018, Vol. 14, no. 2, pp. 217-234
Tsiganov A. V.
A new approach to exact discretization of the Duffing equation is presented. Integrable discrete maps are obtained by using well-studied operations from the elliptic curve cryptography.
Keywords: integrable maps, divisor arithmetic
Citation: Tsiganov A. V., Duffing Oscillator and Elliptic Curve Cryptography, Rus. J. Nonlin. Dyn., 2018, Vol. 14, no. 2, pp. 235-241
Belichenko M. V.
This paper addresses the motion of a Lagrange top in a homogeneous gravitational field under the assumption that the suspension point of the top undergoes high-frequency vibrations with small amplitude in three-dimensional space. The laws of motion of the suspension point are supposed to allow vertical relative equilibria of the top’s symmetry axis. Within the framework of an approximate autonomous system of differential equations of motion written in canonical Hamiltonian form, pendulum-type motions of the top are considered. For these motions, its symmetry axis performs oscillations of pendulum type near the lower, upper or inclined relative equilibrium positions, rotations or asymptotic motions. Integration of the equation of pendulum motion of the top is carried out in the whole range of the problem parameters. The question of their orbital linear stability with respect to spatial perturbations is considered on the isoenergetic level corresponding to the unperturbed motions. The stability evolution of oscillations and rotations of the Lagrange top depending on the ratios between the intensities of the vertical, horizontal longitudinal and horizontal transverse components of vibration is described.
Keywords: Lagrange’s top, high-frequency vibrations, pendulum-type motions, stability
Citation: Belichenko M. V., On the Stability of Pendulum-type Motions in the Approximate Problem of Dynamics of a Lagrange Top with a Vibrating Suspension Point, Rus. J. Nonlin. Dyn., 2018, Vol. 14, no. 2, pp. 243-263
Adamov B. I.
The object of the study is the mobile platform of the KUKA youBot robot equipped with four Mecanum wheels. The ideal conditions for the point contact of the wheels and the floor are considered. It is assumed that the rollers of each Mecanum wheel move without slipping and the center of the wheel, the center of the roller axis, and the point of contact of the roller with the floor are located on the same straight line. The dynamics of the system is described using Appel’s equations and taking into account the linear forces of viscous friction in the joints of the bodies. An algorithm for determination of the control forces is designed. Their structure is the same as that of the reactions of ideal constraints determined by the program motion of the point of the platform. The controlled dynamics of the system is studied using uniform circular motion of the platform point as an example: conditions for the existence and stability of steady rotations are found, conditions for the existence of stable-unstable stationary regimes and rotational motions of the platform are obtained. Within the framework of the theory of singular perturbations, an asymptotic analysis of the rotation of the platform is carried out.
Keywords: omniwheel, Mecanum wheel, omnidirectional platform, servo-constraint, youBot, singular perturbation
Citation: Adamov B. I., A Study of the Controlled Motion of a Four-wheeled Mecanum Platform, Rus. J. Nonlin. Dyn., 2018, Vol. 14, no. 2, pp. 265-290