Vol. 12, No. 4

Vol. 12, No. 4, 2016

Shlufman K. V.,  Neverova G. P.,  Frisman E. Y.
This paper investigates the emergence and stability of 2-cycles for the Ricker model with the 2-year periodic Malthusian parameter. It is shown that the stability loss of the trivial solution occurs through the transcritical bifurcation resulting in a stable 2-cycle. The subsequent tangent bifurcation leads to the appearance of two new 2-cycles: stable and unstable ones. As a result, there is multistability. It is shown that the coexistence of two different stable 2-cycles is possible in a narrow area of the parameter space. Further stability loss of the 2-cycles occurs according to the Feigenbaum scenario.
Keywords: recurrence equation, Ricker model, periodic Malthusian parameter, stability, bifurcation, multistability
Citation: Shlufman K. V.,  Neverova G. P.,  Frisman E. Y., Two-cycles of the Ricker model with the periodic Malthusian parameter: stability and multistability, Rus. J. Nonlin. Dyn., 2016, Vol. 12, No. 4, pp. 553-565
Sysoev I. V.,  Ponomarenko V. I.
Reconstruction of equations of oscillatory systems from time series is an important problem, since results can be useful in different practical applications, including forecast of future dynamics, indirect measurement of parameters and diagnostics of coupling. The problem of reconstruction of coupling coefficients from time series of ensembles of a large number of oscillators is a practically valid problem. This study aims to develop a method of reconstruction of equations of an ensemble of identical neuron-like oscillators in the presence of time delays in couplings based on a given general form of equations.
The proposed method is based on the previously developed approach for reconstruction of diffusively coupled ensembles of time-delayed oscillators. To determine coupling coefficients, the target function is minimized with least-squares routine for each oscillator independently. This function characterizes the continuity of experimental data. Time delays are revealed using a special version of the gradient descent method adapted to the discrete case.
It is shown in the numerical experiment that the proposed method allows one to accurately estimate most of time delays (∼99%) even if short time series are used. The method is asymptotically unbiased.
Keywords: time series, ensembles of oscillators, time delay in coupling, reconstruction of equations
Citation: Sysoev I. V.,  Ponomarenko V. I., Reconstruction of the coupling matrix in the ensemble of identical neuron-like oscillators with time delay in coupling, Rus. J. Nonlin. Dyn., 2016, Vol. 12, No. 4, pp. 567–576
Morozov A. D.,  Morozov K. E.
The two-dimensional nonautonomous equations of pendular type are considered: the Josephson equation and the equation of oscillations of a body. It is supposed that these equations are transitory, i.e., nonautonomous only on a finite time interval. The problem of dependence of the mode on the transitory shift is solved. For a conservative case the measure of transport from oscillations to rotations is established.
Keywords: transitory system, separatrix, limit cycles, attractors
Citation: Morozov A. D.,  Morozov K. E., Transitory shift in pendular type equations, Rus. J. Nonlin. Dyn., 2016, Vol. 12, No. 4, pp. 577–589
Revutskaya O. L.,  Neverova G. P.,  Kulakov M. P.,  Frisman E. Y.
This paper is concerned with the model of dynamics for population with a simple age structure. It is assumed that the growth of population size is regulated by limiting the survival rate of younger individuals. It is shown that the density-dependent regulation of offspring survival can lead to fluctuations in population size. Moreover, there are multistability areas in which the type of dynamic regimes depends on the initial conditions. These aspects of dynamic behavior can explain the changes in the oscillation period, and the appearance and disappearance of population size fluctuations.
Keywords: mathematical modeling, population dynamics, age structure, density-dependent regulation, stability, bifurcations, dynamic modes, multistability, chaos
Citation: Revutskaya O. L.,  Neverova G. P.,  Kulakov M. P.,  Frisman E. Y., Model of age-structured population dynamics: stability, multistability, and chaos, Rus. J. Nonlin. Dyn., 2016, Vol. 12, No. 4, pp. 591–603
Martynov S. I.,  Tkach L. Y.
The model of an aggregate of spherical particles connected by rods and moving in a viscous fluid is considered. The motion of the aggregate is due to the action of hydrodynamic forces from the vortex flow of the viscous fluid. The fluid flow is generated by rotation in opposite directions of two particles of the aggregate. The rotation of the particles is caused by the action of moments of the internal forces whose sum is equal to zero. Other particles of the aggregate are subject to constraints preventing their rotation. To calculate the dynamics of the aggregate, a system of equations of the viscous fluid is jointly solved in the approximation of small Reynolds numbers with appropriate boundary conditions and equations of motion of the particles under the action of applied external and internal forces and torques. The hydrodynamic interaction of the particles is taken into account. It is assumed that the rods do not interact with the fluid and do not allow the particles to change the distance between them. Computer simulation of the dynamics of three different aggregates of 5 particles is tested by special software. The forces in the rods and the speed of movement for each aggregate are calculated. It was found that one aggregate moves faster than others. This means that the shape of the aggregate is more adapted for such movement as compared to the other two. This approach can be used as a basis to create a model of self-propelled aggregates of different geometric shape with two or more pairs of rotating particles. Examples of constructions of aggregates and their dynamics in viscous fluid are also studied by computer simulation.
Keywords: numerical simulation, viscous fluid, particle aggregates, hydrodynamic interaction, internal interaction forces
Citation: Martynov S. I.,  Tkach L. Y., On one model of the dynamics of self-propelled aggregates of particles in a viscous fluid, Rus. J. Nonlin. Dyn., 2016, Vol. 12, No. 4, pp. 605–618
Bardin B. S.,  Chekina E. A.
We deal with the problem of stability for a resonant rotation of a satellite. It is supposed that the satellite is a rigid body whose center of mass moves in an elliptic orbit. The resonant rotation is a planar motion such that the body completes one rotation in absolute space during two orbital revolutions of its center of mass. The stability analysis of the resonant rotation with respect to planar perturbations has been performed in detail earlier. In this paper we investigate the stability of the resonant rotation with respect to both planar and spatial perturbations for a nonsymmetric satellite. For small values of the eccentricity we have obtained boundaries of instability domains (parametric resonance domains) in an analytic form. For arbitrary eccentricity values we numerically construct domains of stability in linear approximation. Outside the above stability domains the resonant rotation is unstable in the sense of Lyapunov.
Keywords: Hamiltonian system, resonant periodic motion, parametric resonance, satellite, stability
Citation: Bardin B. S.,  Chekina E. A., On the stability of a resonant rotation of a satellite in an elliptic orbit, Rus. J. Nonlin. Dyn., 2016, Vol. 12, No. 4, pp. 619–632
Ryabov P. E.,  Biryucheva E.
The paper presents explicitly the spectral curve and the discriminant set of the integrable case of M. Adler and P. van Moerbeke. For critical points of rank 0 and 1 of the momentum map we explicitly calculate the characteristic values defining their type. An algorithm is proposed for finding the bifurcation diagram from the real part of the discriminant set with the help of critical points of rank 0 and 1. The algorithm works under the condition that the real part of the discriminant set contains the bifurcation diagram.
Keywords: integrable Hamiltonian systems, spectral curve, discriminant set, bifurcation diagram
Citation: Ryabov P. E.,  Biryucheva E., The discriminant set and bifurcation diagram of the integrable case of M. Adler and P. van Moerbeke, Rus. J. Nonlin. Dyn., 2016, Vol. 12, No. 4, pp. 633–650
Gorr G. V.
The Bobylev–Steklov solution belongs to one of the most well-known particular solutions of the Euler–Poisson equation of the problem of motion of a heavy rigid body with a fixed point. It is characterized by two linear invariant relations and can be expressed as elliptic functions of time. The interpretation of the motion of the Bobylev–Steklov gyroscope was carried out by P.V. Kharlamov using the Poinsot method. Analysis of the neighborhood of the Bobylev–Steklov solution in the integral manifold of the Euler–Poisson equations was presented by B.S. Bardin for the case where this solution describes pendulum motions. It is therefore of interest to study the general case of the above-mentioned manifold. Using the first Lyapunov method, a new class of asymptotic motions is obtained for a heavy rigid body whose limit motions are described by the Bobylev–Steklov solution.
Keywords: the first Lyapunov method, Bobylev–Steklov solution
Citation: Gorr G. V., On asymptotic motions of a heavy rigid body in the Bobylev–Steklov case, Rus. J. Nonlin. Dyn., 2016, Vol. 12, No. 4, pp. 651–661
Vetchanin E. V.,  Kilin A. A.
This paper is concerned with the motion of an unbalanced heavy three-axial ellipsoid in an ideal fluid controlled by rotation of three internal rotors. It is proved that the motion of the body considered is controlled with respect to configuration variables except for some special cases. An explicit control that makes it possible to implement unbounded motion in an arbitrary direction has been calculated. Directions for which control actions are bounded functions of time have been determined.
Keywords: ideal fluid, motion of a rigid body, Kirchhoff equations, control by rotors, gaits
Citation: Vetchanin E. V.,  Kilin A. A., Control of the motion of an unbalanced heavy ellipsoid in an ideal fluid using rotors, Rus. J. Nonlin. Dyn., 2016, Vol. 12, No. 4, pp. 663–674
Bizyaev I. A.,  Borisov A. V.,  Mamaev I. S.
This paper is concerned with the motion of the Chaplygin sleigh on the surface of a circular cylinder. In the case of inertial motion, the problem reduces to the study of the dynamical system on a (two-dimensional) torus and to the classification of singular points. Particular cases in which the system admits an invariant measure are found. In the case of a balanced and dynamically symmetric Chaplygin sleigh moving in a gravitational field it is shown that on the average the system has no drift along the vertical.
Keywords: Chaplygin sleigh, invariant measure, nonholonomic mechanics
Citation: Bizyaev I. A.,  Borisov A. V.,  Mamaev I. S., Dynamics of the Chaplygin sleigh on a cylinder, Rus. J. Nonlin. Dyn., 2016, Vol. 12, No. 4, pp. 675–687

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