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# Vol. 15, no. 4

 Adabrah A. K.,  Dragović V.,  Radnović M. Abstract We derive necessary and sufficient conditions for periodic trajectories of billiards within an ellipse in the Minkowski plane in terms of an underlying elliptic curve. Equivalent conditions are derived in terms of polynomial-functional equations as well. The corresponding polynomials are related to the classical extremal polynomials. Similarities and differences with respect to the previously studied Euclidean case are indicated. Keywords: Minkowski plane, elliptical billiards, elliptic curve, Akhiezer polynomials Citation: Adabrah A. K.,  Dragović V.,  Radnović M., Elliptical Billiards in the Minkowski Plane and Extremal Polynomials, Rus. J. Nonlin. Dyn., 2019, Vol. 15, no. 4, pp. 397-407 DOI:10.20537/nd190401
 Ardentov A. A. Abstract This investigation continues the study of the classical problem of stationary configurations of an elastic rod on a plane. The length of the rod, the ends of the rod and the directions at the ends are fixed. The problem was first studied by Leonard Euler in 1744 and the optimal synthesis problem is still an open problem. Euler described a family of geodesics containing the solutions, which are called Euler elasticae. It is known that sufficiently small pieces of Euler elasticae are optimal, i.e., they have a minimum of the potential energy. In theory, the point where an optimal curve loses its optimality is called a cut point. Usually several optimal curves arrive at such points, so the points have multiplicity more than 1 and are called Maxwell points. The aim of this work is to describe numerically Maxwell points where two nonsymmetric elasticae come with the same length and energy value. Keywords: Euler elastica, Maxwell strata, optimal control Citation: Ardentov A. A., Hidden Maxwell Stratum in Euler's Elastic Problem, Rus. J. Nonlin. Dyn., 2019, Vol. 15, no. 4, pp. 409-414 DOI:10.20537/nd190402
 Bardin B. S.,  Chekina E. Abstract The motion of a rigid body satellite about its center of mass is considered. The problem of the orbital stability of planar pendulum-like rotations of the satellite is investigated. It is assumed that the satellite moves in a circular orbit and its geometry of mass corresponds to a plate. In unperturbed motion the minor axis of the inertia ellipsoid lies in the orbital plane. A nonlinear analysis of the orbital stability for previously unexplored values of parameters corresponding to the boundaries of the stability regions is carried out. The study is based on the normal form technique. In the special case of fast rotations a normalization procedure is performed analytically. In the general case the coefficients of normal form are calculated numerically. It is shown that in the case under consideration the planar rotations of the satellite are mainly unstable, and only on one of the boundary curves there is a segment where the formal orbital stability takes place. Keywords: satellite, rotations, orbital stability, Hamiltonian system, symplectic map, normal form, combinational resonance, resonance of essential type Citation: Bardin B. S.,  Chekina E., On Orbital Stability of Pendulum-like Satellite Rotations at the Boundaries of Stability Regions, Rus. J. Nonlin. Dyn., 2019, Vol. 15, no. 4, pp. 415-428 DOI:10.20537/nd190403
 Bountis A.,  Kominis Y.,  Shena J.,  Kovanis V. Abstract Coupled semiconductor lasers are systems possessing complex dynamics, which makes them interesting for many applications in photonics. In this paper, we first review our results on the existence and stability of asymmetric phase-locked states of a single dimer consisting of two coupled semiconductor lasers. We show that stable phase-locked states of arbitrary asymmetry exist, whose field amplitude ratio and phase difference can be dynamically controlled by appropriate electronic current injection. Moreover, we obtain stable limit cycles with asymmetric characteristics, emerging through Hopf bifurcations from these phase-locked states. Also, we emphasize the importance of exceptional points, and we show that asymmetry enables their existence in extended regions of parameter space. The dynamics of asymmetric dimers under small signal modulation of the pumping current is also investigated and the occurrence of antiresonances and sharp resonances with very high frequencies is demonstrated. Finally, we describe our recent findings on optically coupled arrays of coupled dimers and explore their fascinating nonlinear dynamics. In particular, we couple in an appropriate way a large number of dimers and show that, depending on their degree of asymmetry, they exhibit organized high amplitude oscillations, or oscillate very close to phase-locked states, suggesting that such photonic networks may prove useful in a variety of beam forming and beam shaping applications. Keywords: coupled mode equations, semiconductor laser arrays, Hopf bifurcations, limit cycle oscillations, coherence in photonic arrays Citation: Bountis A.,  Kominis Y.,  Shena J.,  Kovanis V., Complex Dynamics Induced by Asymmetry in Coupled Laser Systems, Rus. J. Nonlin. Dyn., 2019, Vol. 15, no. 4, pp. 429-455 DOI:10.20537/nd190404
 Gajić B.,  Jovanović B. Abstract We consider the nonholonomic problem of rolling without slipping and twisting of a balanced ball over a fixed sphere in $\mathbb{R}^n$. By relating the system to a modified LR system, we prove that the problem always has an invariant measure. Moreover, this is a~$SO(n)$-Chaplygin system that reduces to the cotangent bundle $T^*S^{n-1}$. We present two integrable cases. The first one is obtained for a special inertia operator that allows the Chaplygin Hamiltonization of the reduced system. In the second case, we consider the rigid body inertia operator $\mathbb I\omega=I\omega+\omega I$, ${I=diag(I_1,\ldots,I_n)}$ with a symmetry $I_1=I_2=\ldots=I_{r} \ne I_{r+1}=I_{r+2}=\ldots=I_n$. It is shown that general trajectories are quasi-periodic, while for $r\ne 1$, $n-1$ the Chaplygin reducing multiplier method does not apply. Keywords: nonholonomic Chaplygin systems, invariant measure, integrability Citation: Gajić B.,  Jovanović B., Two Integrable Cases of a Ball Rolling over a Sphere in $\mathbb{R}^n$, Rus. J. Nonlin. Dyn., 2019, Vol. 15, no. 4, pp. 457-475 DOI:10.20537/nd190405
 Golousov S. V.,  Khusainov R. R.,  Savin S. I. Abstract The paper deals with one of the modern challenges in walking robotics: moving across a rough terrain where the geometry of the terrain is unknown and hence it is impossible to plan precise trajectories for the robot feet in advance, before a collision with the supporting surface occurs. In this paper, an algorithm for the dynamics correction of the foot trajectory based on the compliant control is employed to deal with the problem. Additionally, to solve the problem of dynamic correction of the foot trajectory, it also provides a biomorphic reaction force profile, which might be a desired property for some applications. Keywords: walking robot, uneven terrain, compliant control, biomorphic reaction force profile Citation: Golousov S. V.,  Khusainov R. R.,  Savin S. I., Compliant Control for Walking Robots with the Use of a Virtual Spring-Damper System, Rus. J. Nonlin. Dyn., 2019, Vol. 15, no. 4, pp. 477-485 DOI:10.20537/nd190406
 Karavaev Y. L.,  Shestakov V. A.,  Yefremov K. S. Abstract This paper presents experimental investigations of the control algorithm of a highly maneuverable mobile manipulation robot. The kinematics of a mobile manipulation robot, the algorithm of trajectory planning of the mobile robot to the point of object gripping are considered. By realization of the algorithm, the following tasks are solved: solution of the inverse positional task for the mobile manipulation robot; motion planning of the mobile manipulator taking into account the minimization of energy and time consumption per movement. The result of the algorithm is a movement to the point of gripping of the manipulation object; grasping and loading of the object. Experimental investigations of the developed algorithms are given. Keywords: mobile manipulation robot, motion planning, trajectory discretization, Kinect Citation: Karavaev Y. L.,  Shestakov V. A.,  Yefremov K. S., Experimental Investigations of the Control Algorithm of a Mobile Manipulation Robot, Rus. J. Nonlin. Dyn., 2019, Vol. 15, no. 4, pp. 487-495 DOI:10.20537/nd190407
 Karavaev Y. L.,  Kilin A. A. Abstract This paper presents the results of the study of the dynamics of a real spherical robot of combined type in the case of control using small periodic oscillations. The spherical robot is set in motion by controlled change of the position of the center of mass and by generating variable gyrostatic momentum. We demonstrate how to use small periodic controls for stabilization of the spherical robot during motion. The results of numerical simulation are obtained for various initial conditions and control parameters that ensure a change in the position of the center of mass and a variation of gyrostatic momentum. The problem of the motion of a spherical robot of combined type on a surface that performs flat periodic oscillations is also considered. The results of numerical simulation are obtained for different initial conditions, control actions and parameters of oscillations. Keywords: spherical robot, nonholonomic constraint, small periodic control actions, stabilization Citation: Karavaev Y. L.,  Kilin A. A., The Dynamics of a Spherical Robot of Combined Type by Periodic Control Actions, Rus. J. Nonlin. Dyn., 2019, Vol. 15, no. 4, pp. 497-504 DOI:10.20537/nd190408
 Kholostova O. V. Abstract The motion of a solid (satellite) carrying a moving point mass in the central Newtonian gravitational field in an elliptical orbit of arbitrary eccentricity is considered. The law of motion of a point mass is assumed to allow for the existence of relative equilibria of the “body-point” system in the orbital coordinate system. A nonlinear stability analysis of these equilibria is carried out, based on the construction and normalization of the area-preserving mapping generated by the motions of the system. Keywords: solid carrying a point mass, elliptical orbit, relative equilibrium, stability, resonance Citation: Kholostova O. V., Nonlinear Stability Analysis of Relative Equilibria of a Solid Carrying a Movable Point Mass in the Central Gravitational Field, Rus. J. Nonlin. Dyn., 2019, Vol. 15, no. 4, pp. 505-512 DOI:10.20537/nd190409
 Kozlov V. V. Abstract It is well known that the maximal value of the central moment of inertia of a closed homogeneous thread of fixed length is achieved on a curve in the form of a circle. This isoperimetric property plays a key role in investigating the stability of stationary motions of a flexible thread. A discrete variant of the isoperimetric inequality, when the mass of the thread is concentrated in a finite number of material particles, is established. An analog of the isoperimetric inequality for an inhomogeneous thread is proved. Keywords: moment of inertia, Sundman and Wirtinger inequalities, articulated polygon Citation: Kozlov V. V., Isoperimetric Inequalities for Moments of Inertia and Stability of Stationary Motions of a Flexible Thread, Rus. J. Nonlin. Dyn., 2019, Vol. 15, no. 4, pp. 513-523 DOI:10.20537/nd190410
 Kurakin L. G.,  Kurdoglyan A. V. Abstract The systems of differential equations with one cosymmetry are considered [1]. The ordinary object for such systems is a one-dimensional continuous family of equilibria. The stability spectrum changes along this family, but it necessarily contains zero. We consider the nondegeneracy condition, thus the boundary equilibria separate the family on linearly stable and instable areas. The stability of the boundary equilibria depends on nonlinear terms of the system. The stability problem for the systems with one cosymmetry is studied in [2]. The general problem is to apply the stability criteria one needs to compute coefficients of the model system. It is especially difficult if the system has a large dimension, while a number of critical variables may be small. A method for calculating coefficients is proposed in [3]. In this work the expressions for the known stability criteria are proposed in a form convenient for calculation. The explicit formulas of the coefficients of the model system are given in semi-invariant form. They are expressed using the generalized eigenvectors of the linear matrix and its conjugate matrix. Keywords: stability, critical case, neutral manifold, cosymmetry, semi-invariant form Citation: Kurakin L. G.,  Kurdoglyan A. V., Semi-Invariant Form of Equilibrium Stability Criteria for Systems with One Cosymmetry, Rus. J. Nonlin. Dyn., 2019, Vol. 15, no. 4, pp. 525-531 DOI:10.20537/nd190411
 Kurakin L. G.,  Ostrovskaya I. V. Abstract In this paper the two-layer geostrophic model of the rotating fluid and the model of Bessel vortices are considered. Kirchhoff's model of vortices in a homogeneous fluid is the limiting case of both of these models. Part of the study is performed for an arbitrary Hamiltonian depending on the distances between point vortices. The review of the stability problem of stationary rotation of regular Thomson's vortex $N$-gon of identical vortices is given for ${N\geqslant 2}$. The stability problem of the vortex tripole/quadrupole is also considered. This axisymmetric vortex structure consists of a~central vortex of an arbitrary intensity and two/three identical peripheral vortices. In the model of a two-layer fluid, peripheral vortices belong to one of the layers and the central vortex can belong to either another layer or the same. The stability of the stationary rotation is interpreted as orbital stability (the stability of a one-parameter orbit of a stationary rotation of a vortex system). The instability of the stationary rotation is instability of equilibrium of the reduced system. The quadratic part of the Hamiltonian and eigenvalues of the linearization matrix are studied. The parameter space is divided into three parts: $\bf A$ is the domain of stability in an exact nonlinear setting, $\bf B$ is the linear stability domain, where the stability problem requires nonlinear analysis, and $\bf C$ is the instability domain. In the stability problem of a vortex multipole, another definition of stability is used; it is the stability of an invariant three-parametric set of all trajectories of the families of stationary orbits. It is shown that in the case of non zero total intensity, the stability of the invariant set implies orbital stability. Keywords: $N$-vortex problem, Thomson's vortex $N$-gon, point vortices, two-layer fluid, stability, Hamiltonian equation Citation: Kurakin L. G.,  Ostrovskaya I. V., On the Stability of Thomson’s Vortex $N$-gon and a Vortex Tripole/Quadrupole in Geostrophic Models of Bessel Vortices and in a Two-Layer Rotating Fluid: a Review, Rus. J. Nonlin. Dyn., 2019, Vol. 15, no. 4, pp. 533-542 DOI:10.20537/nd190412
 Kuzenov V. V.,  Ryzhkov S. V. Abstract A statement of the problem is presented and numerical modeling of plasma-gas-dynamic processes in the capillary discharge plume is performed. In the developed model, plasma dynamic processes in a capillary discharge are determined by the intensity, duration of plasma formation processes in the capillary discharge channel, and thermodynamic parameters in the surrounding gaseous medium. The spatial distribution of temperature, density and pressure, radial and longitudinal velocities of pulsed jets of several capillary discharge channels is presented. Keywords: capillary discharge, numerical method, plasma dynamics Citation: Kuzenov V. V.,  Ryzhkov S. V., Mathematical Modeling of Plasma Dynamics for Processes in Capillary Discharges, Rus. J. Nonlin. Dyn., 2019, Vol. 15, no. 4, pp. 543-550 DOI:10.20537/nd190413
 Kuznetsov S. P. Abstract The article considers the Chaplygin sleigh on a plane in a potential well, assuming that an external potential force is supplied at the mass center. Two particular cases are studied in some detail, namely, a one-dimensional potential valley and a potential with rotational symmetry; in both cases the models reduce to four-dimensional differential equations conserving mechanical energy. Assuming the potential functions to be quadratic, various behaviors are observed numerically depending on the energy, from those characteristic to conservative dynamics (regularity islands and chaotic sea) to strange attractors. This is another example of a nonholonomic system manifesting these phenomena (similar to those for Celtic stone or Chaplygin top), which reflects a fundamental nature of these systems occupying an intermediate position between conservative and dissipative dynamics. Keywords: Chaplygin sleigh, nonholonomic system, chaos, attractor Citation: Kuznetsov S. P., Complex Dynamics in Generalizations of the Chaplygin Sleigh, Rus. J. Nonlin. Dyn., 2019, Vol. 15, no. 4, pp. 551-559 DOI:10.20537/nd190414
 Mashtakov A. P. Abstract This paper summarizes results of a sequence of works related to usage of sub-Riemannian (SR) geometry in image processing and modeling of the human visual system. In recent research in psychology of vision (J. Petitot, G.Citti, A. Sarti) it was shown that SR geodesics appear as natural curves that model a mechanism of the primary visual cortex V1 of a human brain for completion of contours that are partially corrupted or hidden from observation. We extend the model to include data adaptivity via a suitable external cost in the SR metric. We show that data adaptive SR geodesics are useful in real image analysis applications and provide a refined model of V1 that takes into account the presence of a visual stimulus. Keywords: sub-Riemannian, detection of salient lines, vision, visual cortex, brain-inspired methods Citation: Mashtakov A. P., Sub-Riemannian Geometry in Image Processing and Modeling of the Human Visual System, Rus. J. Nonlin. Dyn., 2019, Vol. 15, no. 4, pp. 561-568 DOI:10.20537/nd190415
 Podobryaev A. Abstract We consider left-invariant optimal control problems on connected Lie groups. We describe the symmetries of the exponential map that are induced by the symmetries of the vertical part of the Hamiltonian system of the Pontryagin maximum principle. These symmetries play a key role in investigation of optimality of extremal trajectories. For connected Lie groups such that the generic coadjoint orbit has codimension not more than 1 and a connected stabilizer we introduce a general construction for such symmetries of the exponential map. Keywords: symmetry, geometric control theory, Riemannian geometry, sub-Riemannian geometry Citation: Podobryaev A., Symmetric Extremal Trajectories in Left-Invariant Optimal Control Problems, Rus. J. Nonlin. Dyn., 2019, Vol. 15, no. 4, pp. 569-575 DOI:10.20537/nd190416
 Sachkov Y. L.,  Sachkova E. F. Abstract The left-invariant sub-Riemannian problem with the growth vector (2, 3, 5, 8) is considered. A two-parameter group of infinitesimal symmetries consisting of rotations and dilations is described. The abnormal geodesic flow is factorized modulo the group of symmetries. A parameterization of the vertical part of abnormal geodesic flow is obtained. Keywords: sub-Riemannian geometry, abnormal extremals, symmetries Citation: Sachkov Y. L.,  Sachkova E. F., Symmetries and Parameterization of Abnormal Extremals in the Sub-Riemannian Problem with the Growth Vector (2, 3, 5, 8), Rus. J. Nonlin. Dyn., 2019, Vol. 15, no. 4, pp. 577-585 DOI:10.20537/nd190417
 Sakharov  A. V. Abstract The problem of three-dimensional motion of a passively gravitating point in the potential created by a homogeneous thin fixed ring and a point located in the center of the ring is considered. Motion of the point allows two first integrals. In the paper equilibrium points and invariant manifolds of the phase space of the system are found. Motions in them are analyzed. Bifurcations in the phase plane corresponding to the motion in the equatorial plane are shown. Keywords: celestial mechanics, axisymmetric potential, center, ring, phase portrait, phase space, first integrals, bifurcations Citation: Sakharov  A. V., Some Trajectories of a Point in the Potential of a Fixed Ring and Center, Rus. J. Nonlin. Dyn., 2019, Vol. 15, no. 4, pp. 587-592 DOI:10.20537/nd190418
 Sukhov E. A. Abstract We deal with motions of a dynamically symmetric rigid-body satellite about its center of mass in a central Newtonian gravitational field. In this case the equations of motion possess particular solutions representing the so-called regular precessions: cylindrical, conical and hyperboloidal precession. If a regular precession is stable there exist two types of periodic motions in its neighborhood: short-periodic motions with a period close to $2\pi / \omega_2$ and long-periodic motions with a~period close to $2 \pi / \omega_1$ where $\omega_2$ and $\omega_1$ are the frequencies of the linearized system ($\omega_2 > \omega_1$). In this work we obtain analytically and numerically families of short-periodic motions arising from regular precessions of a symmetric satellite in a nonresonant case and long-periodic motions arising from hyperboloidal precession in cases of third- and fourth-order resonances. We investigate the bifurcation problem for these families of periodic motions and present the results in the form of bifurcation diagrams and Poincaré maps. Keywords: Hamiltonian mechanics, satellite dynamics, bifurcations, periodic motions, orbital stability Citation: Sukhov E. A., Bifurcation Analysis of Periodic Motions Originating from Regular Precessions of a Dynamically Symmetric Satellite, Rus. J. Nonlin. Dyn., 2019, Vol. 15, no. 4, pp. 593-609 DOI:10.20537/nd190419
 Tafrishi S. A.,  Bai Y.,  Svinin M.,  Esmaeilzadeh E.,  Yamamoto M. Abstract In this paper, the rest-to-rest motion planning problem of a fluid-actuated spherical robot is studied. The robot is driven by moving a spherical mass within a circular fluid-filled pipe fixed internally to the spherical shell. A mathematical model of the robot is established and two inverse dynamics-based feed-forward control methods are proposed. They parameterize the motion of the outer shell or the internal moving mass as weighted Beta functions. The feasibility of the proposed feed-forward control schemes is verified under simulations. Keywords: inverse dynamics, motion planning, spherical robots, fluid actuator Citation: Tafrishi S. A.,  Bai Y.,  Svinin M.,  Esmaeilzadeh E.,  Yamamoto M., Inverse Dynamics-Based Motion Control of a Fluid-Actuated Rolling Robot, Rus. J. Nonlin. Dyn., 2019, Vol. 15, no. 4, pp. 611-622 DOI:10.20537/nd190420
 Vorochaeva (Volkova) L. Y.,  Savin S. I.,  Yatsun A. S. Abstract This paper deals with the gait of a robot crawling on a horizontal rough surface with dry friction. A distinctive feature of the device is the presence of two supporting elements with a controlled coefficient of friction, which allows for alternately fixing the supports on the surface. As a result of numerical simulation, the patterns (laws) of influence on the motion characteristics of the mass-dimensional and control parameters of the robot, as well as the parameters of the supporting surface, are obtained, so that it is possible to find application in the design of the specified devices. Keywords: crawling robot, supporting element, controlled friction coefficient, dry friction model Citation: Vorochaeva (Volkova) L. Y.,  Savin S. I.,  Yatsun A. S., An Investigation of Motion of a Crawling Robot with Supports with Controllable Friction, Rus. J. Nonlin. Dyn., 2019, Vol. 15, no. 4, pp. 623-632 DOI:10.20537/nd190421