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# Vol. 16, no. 2, 2020

 Yehia H. M.,  Hussein A. M. Abstract The method introduced in [11] and [12] is extended to construct new families of severalparameter integrable systems, which admit a complementary integral quartic in the velocities. A list of 14 systems is obtained, of which 12 are new. Each of the new systems involves a number of parameters ranging from 7 up to 16 parameters entering into its structures. A detailed preliminary analysis of certain special cases of one of the new systems is performed, aimed at obtaining some global results. We point out twelve combinations of conditions on the parameters which characterize integrable dynamics on Riemannian manifolds as configuration spaces. Very special 7 versions of the 12 cases are interpreted as new integrable motions with a quartic integral in the Poincaré half-plane. A byproduct of the process of solution is the construction of 12 Riemannian metrics whose geodesic flow is integrable with a quartic second integral. Keywords: integrable systems, quartic second integrals, Poincaré half-plane Citation: Yehia H. M.,  Hussein A. M., New Families of Integrable Two-Dimensional Systems with Quartic Second Integrals, Rus. J. Nonlin. Dyn., 2020, Vol. 16, no. 2, pp. 211-242 DOI:10.20537/nd200201
 Bukh A. V.,  Anishchenko V. S. Abstract The features of external and mutual synchronization of spiral wave structures including chimera states in the interacting two-dimensional lattices of nonlocally coupled Nekorkin maps are investigated. The cases of diffusive and inertial couplings between the lattices are considered. The lattices model a neuronal activity and represent two-dimensional lattices consisting of $N \times N$ elements with $N = 200$. It is shown that the effect of complete synchronization is not achieved in the studied lattices, and only the regime of partial synchronization is realized regardless of the case of coupling between the lattices. It is important to note that the conclusion is applied not only to the regimes of spiral wave chimeras, but also to the regimes of regular spiral waves. Keywords: synchronization, two-dimensional lattice, spiral wave, spiral wave chimera, inertial and diffusing coupling Citation: Bukh A. V.,  Anishchenko V. S., Features of the Synchronization of Spiral Wave Structures in Interacting Lattices of Nonlocally Coupled Maps, Rus. J. Nonlin. Dyn., 2020, Vol. 16, no. 2, pp. 243-257 DOI:10.20537/nd200202
 Burov A. A.,  Nikonov V. I. Abstract As is well known, many small celestial bodies are of a rather complex shape. Therefore, the study of the dynamics of a spacecraft in their vicinity, based on terms up to the second order of smallness in the expansion of the potential of attraction, seems to be insufficient for an adequate description of the observed dynamical effects related, for example, to positioning of the libration points. In this paper, such effects are demonstrated for spacecraft dynamics in the vicinity of the asteroid (2063) Bacchus. The libration points are computed for various approximations of the gravitational potential. The results of this computation are compared with similar results obtained before for the so-called Sludsky – Werner – Scheeres potential. The dependence of the structure of the regions of possible motions on approximation of the gravitational potential is also studied. Keywords: (2063) Bacchus, gravitational potential expansion, libration points, region of possible motion, Hill’s region, zero-velocity locus Citation: Burov A. A.,  Nikonov V. I., Inertial Characteristics of Higher Orders and Dynamics in a Proximity of a Small Celestial Body, Rus. J. Nonlin. Dyn., 2020, Vol. 16, no. 2, pp. 259-273 DOI:10.20537/nd200203
 Mogilevich L.,  Ivanov S.,  Blinkov Y. Abstract This article investigates longitudinal deformation waves in physically nonlinear coaxial elastic shells containing a viscous incompressible fluid between them. The rigid nonlinearity of the shells is considered. The presence of a viscous incompressible fluid between the shells, as well as the influence of the inertia of the fluid motion on the amplitude and velocity of the wave, are taken into account. A numerical study of the model constructed in the course of this work is carried out by using a difference scheme for the equation similar to the Crank – Nicolson scheme for the heat equation. In the case of identical initial conditions in both shells, the deformation waves in them do not change either the amplitude or the velocity. In the case of setting different initial conditions in the coaxial shells, the amplitude of the solitary wave in the first shell decreases from the value specified at the initial instant of time, and in the second, the amplitude grows from zero until they equalize, that is, energy is transferred. The movement occurs in a negative direction. This means that the velocity of deformation wave is subsonic. Keywords: nonlinear waves, elastic cylindrical shells, viscous incompressible fluid, Crank – Nicolson difference scheme Citation: Mogilevich L.,  Ivanov S.,  Blinkov Y., Modeling of Nonlinear Waves in Two Coaxial Physically Nonlinear Shells with a Viscous Incompressible Fluid Between Them, Taking into Account the Inertia of its Motion, Rus. J. Nonlin. Dyn., 2020, Vol. 16, no. 2, pp. 275-290 DOI:10.20537/nd200204
 Adamov B. I.,  Saypulaev G. R. Abstract The subject of this study is an omnidirectional mobile platform equipped with four Mecanum wheels. The movement of the system on a horizontal plane is considered. The aim of this research is to study the dynamics of the omnidirectional platform, taking into account the design of Mecanum wheels: the shape of the rollers and their finite number. The equations of motion of the onmidirectional mobile platform are derived taking into account the real design of the Mecanum wheels and their slippage. A comparative analysis of the results of numerical modeling for different models of contact friction forces is presented. It has been established that switching of contact rollers and displacement of contact points lead to the occurrence of high-frequency components of wheel rotation speeds, as well as an offset of their average values (in comparison with the modeling results without taking into account the design features of the chassis). Keywords: omniplatform, Mecanum platform, Mecanum wheel, youBot, omniwheel Citation: Adamov B. I.,  Saypulaev G. R., Research on the Dynamics of an Omnidirectional Platform Taking into Account Real Design of Mecanum Wheels (as Exemplified by KUKA youBot), Rus. J. Nonlin. Dyn., 2020, Vol. 16, no. 2, pp. 291-307 DOI:10.20537/nd200205
 Essaouini H.,  Capodanno P. Abstract We study the small oscillations of a pendulum completely filled by a viscoelastic fluid, restricting ourselves for the fluid to the simpler Oldroyd model. We establish the equations of motion of the system. Writing them in a suitable form, we obtain an existence and unicity theorem of the solution of the associated evolution problem by means of semigroup theory. Afterwards, we show the existence and symmetry of the spectrum and prove the stability of the system. We show the existence of two sets of positive real eigenvalues, of which the first has infinity, and the second a point of the real axis, as points of accumulation. Finally, we specify the location of the possible nonreal eigenvalues. Keywords: viscoelastic fluid, small oscillations, variational-operatorial and spectral methods, semigroups Citation: Essaouini H.,  Capodanno P., Mathematical Study of the Small Oscillations of a Pendulum Completely Filled with a Viscoelastic Fluid, Rus. J. Nonlin. Dyn., 2020, Vol. 16, no. 2, pp. 309-324 DOI:10.20537/nd200206
 Kuzenov V. V.,  Ryzhkov S. V.,  Starostin A. V. Abstract This work is devoted to the theoretical calculation of the processes of compression and energy release in the target by a combined action of a system of pulsed jets and intense laser radiation using a magnetic inertial plasma confinement method. A mathematical model, a numerical method, and a computational algorithm are developed to describe plasma-physical processes occurring in various types of high-temperature installations with high density. The results of the calculation of the hybrid effect of intensive energy flows on a cylindrical target are presented. The main gas-dynamic and radiative parameters of the compressed target plasma are found. Keywords: computer simulation, magnetized target, mathematical modeling, numerical algorithm Citation: Kuzenov V. V.,  Ryzhkov S. V.,  Starostin A. V., Development of a Mathematical Model and the Numerical Solution Method in a Combined Impact Scheme for MIF Target, Rus. J. Nonlin. Dyn., 2020, Vol. 16, no. 2, pp. 325-341 DOI:10.20537/nd200207
 Polekhin I. Y. Abstract In this paper we study the existence of forced oscillations in two Lagrange systems with gyroscopic forces: a spherical pendulum in a magnetic field and a point on a rotating closed convex surface. We show how it is possible to prove the existence of forced oscillations in these systems provided the systems move in the presence of viscous friction. Keywords: forced oscillation, spherical pendulum, gyroscopic force, friction, Wa˙zewski method Citation: Polekhin I. Y., Remarks on Forced Oscillations in Some Systems with Gyroscopic Forces, Rus. J. Nonlin. Dyn., 2020, Vol. 16, no. 2, pp. 343-353 DOI:10.20537/nd200208
 Sachkov Y. L. Abstract The Engel group is the four-dimensional nilpotent Lie group of step 3, with 2 generators. We consider a one-parameter family of left-invariant rank 2 sub-Finsler problems on the Engel group with the set of control parameters given by a square centered at the origin and rotated by an arbitrary angle. We adopt the viewpoint of time-optimal control theory. By Pontryagin’s maximum principle, all sub-Finsler length minimizers belong to one of the following types: abnormal, bang-bang, singular, and mixed. Bang-bang controls are piecewise controls with values in the vertices of the set of control parameters. We describe the phase portrait for bang-bang extremals. In previous work, it was shown that bang-bang trajectories with low values of the energy integral are optimal for arbitrarily large times. For optimal bang-bang trajectories with high values of the energy integral, a general upper bound on the number of switchings was obtained. In this paper we improve the bounds on the number of switchings on optimal bang-bang trajectories via a second-order necessary optimality condition due to A. Agrachev and R.Gamkrelidze. This optimality condition provides a quadratic form, whose sign-definiteness is related to optimality of bang-bang trajectories. For each pattern of these trajectories, we compute the maximum number of switchings of optimal control. We show that optimal bang-bang controls may have not more than 9 switchings. For particular patterns of bang-bang controls, we obtain better bounds. In such a way we improve the bounds obtained in previous work. On the basis of the results of this work we can start to study the cut time along bang-bang trajectories, i.e., the time when these trajectories lose their optimality. This question will be considered in subsequent work. Keywords: sub-Finsler problem, Engel group, bang-bang extremal, optimality condition Citation: Sachkov Y. L., Optimal Bang-Bang Trajectories in Sub-Finsler Problems on the Engel Group, Rus. J. Nonlin. Dyn., 2020, Vol. 16, no. 2, pp. 355-367 DOI:10.20537/nd200209
 Morozov A. D.,  Morozov K. E. Abstract We study nonconservative quasi-periodic $m$-frequency $\it parametric$ perturbations of twodimensional nonlinear Hamiltonian systems. Our objective is to specify the conditions for the existence of new regimes in resonance zones, which may arise due to parametric terms in the perturbation. These regimes correspond to $(m + 1)$-frequency quasi-periodic solutions, which are not generated from Kolmogorov tori of the unperturbed system. The conditions for the existence of these solutions are found. The study is based on averaging theory and the analysis of the corresponding averaged systems. We illustrate the results with an example of a Duffing type equation. Keywords: resonances, quasi-periodic, parametric, averaging method, limit cycles, invariant torus, phase curves, equilibrium states Citation: Morozov A. D.,  Morozov K. E., On Quasi-Periodic Parametric Perturbations of Hamiltonian Systems, Rus. J. Nonlin. Dyn., 2020, Vol. 16, no. 2, pp. 369-378 DOI:10.20537/nd200210
 Kugushev E. I.,  Popova T. V. Abstract We consider systems of ordinary differential equations whose right-hand sides contain timeperiodic functions with some frequencies. An averaged system is constructed by introduction of additional variables and by step-by-step averaging over these variables. An upper estimate of the deviation of the solution to the initial system from the solution to the averaged system is given. Examples are given of mechanical systems in which vibrations with several frequencies occur and for the analysis of which the statements obtained are applied. Keywords: averaging method, multifrequency perturbations, vibration frequency Citation: Kugushev E. I.,  Popova T. V., Estimation of the Accuracy of the Averaging Method for Systems with Multifrequency Perturbations, Rus. J. Nonlin. Dyn., 2020, Vol. 16, no. 2, pp. 379-394 DOI:10.20537/nd200211

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