 EDITORINCHIEF
 DEPUTY EDITORINCHIEF
 Editorial board

Volume 21, Number 78
Nonlinear Dynamics & Mobile Robotics
Ardentov A. A.
Controlling of a Mobile Robot with a Trailer and Its Nilpotent Approximation
Abstract
This work studies a number of approaches to solving the motion planning problem for a mobile robot with a trailer. Different control models of carlike robots are considered from the differentialgeometric point of view. The same models can also be used for controlling a mobile robot with a trailer. However, in cases where the position of the trailer is of importance, i.e., when it is moving backward, a more complex approach should be applied. At the end of the article, such an approach, based on recent works in subRiemannian geometry, is described. It is applied to the problem of reparking a trailer and implemented in the algorithm for parking a mobile robot with a trailer.

Borisov A. V., Kuznetsov S. P.
Regular and Chaotic Motions of a Chaplygin Sleigh under Periodic Pulsed Torque Impacts
Abstract
For a Chaplygin sleigh on a plane, which is a paradigmatic system of nonholonomic mechanics, we consider dynamics driven by periodic pulses of supplied torque depending on the instant spatial orientation of the sleigh. Additionally, we assume that a weak viscous force and moment affect the sleigh in time intervals between the pulses to provide sustained modes of the motion associated with attractors in the reduced threedimensional phase space (velocity, angular velocity, rotation angle). The developed discrete version of the problem of the Chaplygin sleigh is an analog of the classical Chirikov map appropriate for the nonholonomic situation. We demonstrate numerically, discuss and classify dynamical regimes depending on the parameters, including regular motions and diffusivelike random walks associated, respectively, with regular and chaotic attractors in the reduced momentum dynamical equations.

Ivanov A. P.
On Final Motions of a Chaplygin Ball on a Rough Plane
Abstract
A heavy balanced nonhomogeneous ball moving on a rough horizontal plane is considered. The classical model of a “marble” body means a single point of contact, where sliding is impossible. We suggest that the contact forces be described by Coulomb’s law and show that in the final motion there is no sliding. Another, relatively new, contact model is the “rubber” ball: there is no sliding and no spinning. We treat this situation by applying a local Coulomb law within a small contact area. It is proved that the final motion of a ball with such friction is the motion of the “rubber” ball.

Kashchenko S. A.
The Dynamics of Secondorder Equations with Delayed Feedback and a Large Coefficient of Delayed Control
Abstract
The dynamics of secondorder equations with nonlinear delayed feedback and a large coefficient of a delayed equation is investigated using asymptotic methods. Based on special methods of quasinormal forms, a new construction is elaborated for obtaining the main terms of asymptotic expansions of asymptotic residual solutions. It is shown that the dynamical properties of the above equations are determined mostly by the behavior of the solutions of some special families of parabolic boundary value problems. A comparative analysis of the dynamics of equations with the delayed feedback of three types is carried out.

Kozlov V. V.
On the Extendability of Noether’s Integrals for Orbifolds of Constant Negative Curvature
Abstract
This paper is concerned with the problem of the integrable behavior of geodesics on homogeneous factors of the Lobachevsky plane with respect to Fuchsian groups (orbifolds). Locally the geodesic equations admit three independent Noether integrals linear in velocities (energy is a quadratic form of these integrals). However, when passing along closed cycles the Noether integrals undergo a linear substitution. Thus, the problem of integrability reduces to the search for functions that are invariant under these substitutions. If a Fuchsian group is Abelian, then there is a first integral linear in the velocity (and independent of the energy integral). Conversely, if a Fuchsian group contains noncommuting hyperbolic or parabolic elements, then the geodesic flow does not admit additional integrals in the form of a rational function of Noether integrals. We stress that this result holds also for noncompact orbifolds, when there is no ergodicity of the geodesic flow (since nonrecurrent geodesics can form a set of positive measure).

Mashtakov A. P., Ardentov A. A., Sachkov Y. L.
Relation Between Euler’s Elasticae and SubRiemannian Geodesics on $SE(2)$
Abstract
In this note we describe a relation between Euler’s elasticae and subRiemannian geodesics on $SE(2)$. Analyzing the Hamiltonian system of the Pontryagin maximum principle, we show that these two curves coincide only in the case when they are segments of a straight line.

Pankratova E. V., Kalyakulina A. I.
Environmentally Induced Amplitude Death and Firing Provocation in Largescale Networks of Neuronal Systems
Abstract
We study the dynamics of multielement neuronal systems taking into account both the direct interaction between the cells via linear coupling and nondiffusive celltocell communication via common environment. For the cells exhibiting individual bursting behavior, we have revealed the dependence of the network activity on its scale. Particularly, we show that smallscale networks demonstrate the inability to maintain complicated oscillations: for a small number of elements in an ensemble, the phenomenon of amplitude death is observed. The existence of threshold network scales and mechanisms causing firing in artificial and real multielement neural networks, as well as their significance for biological applications, are discussed.

Smirnov L. A., Kryukov A. K., Osipov G. V., Kurths J.
Bistability of Rotational Modes in a System of Coupled Pendulums
Abstract
The main goal of this research is to examine any peculiarities and special modes observed in the dynamics of a system of two nonlinearly coupled pendulums. In addition to steady states, an inphase rotation limit cycle is proved to exist in the system with both damping and constant external force. This rotation mode is numerically shown to become unstable for certain values of the coupling strength. We also present an asymptotic theory developed for an infinitely small dissipation, which explains why the inphase rotation limit cycle loses its stability. Boundaries of the instability domain mentioned above are found analytically. As a result of numerical studies, a whole range of the coupling parameter values is found for the case where the system has more than one rotation limit cycle. There exist not only a stable inphase cycle, but also two outof phase ones: a stable rotation limit cycle and an unstable one. Bistability of the limit periodic mode is, therefore, established for the system of two nonlinearly coupled pendulums. Bifurcations that lead to the appearance and disappearance of the outofphase limit regimes are discussed as well.

Anastassiou S.
Dynamical Systems on the Liouville Plane and the Related Strictly Contact Systems
Abstract
We study vector fields of the plane preserving the Liouville form. We present their local models up to the natural equivalence relation and describe local bifurcations of low codimension. To achieve that, a classification of univariate functions is given according to a relation stricter than contact equivalence. In addition, we discuss their relation with strictly contact vector fields in dimension three. Analogous results for diffeomorphisms are also given.

Vetchanin E. V., Kilin A. A., Mamaev I. S.
Control of the Motion of a Helical Body in a Fluid Using Rotors
Abstract
This paper is concerned with the motion of a helical body in an ideal fluid, which is controlled by rotating three internal rotors. It is proved that the motion of the body is always controllable by means of three rotors with noncoplanar axes of rotation. A condition whose satisfaction prevents controllability by means of two rotors is found. Control actions that allow the implementation of unbounded motion in an arbitrary direction are constructed. Conditions under which the motion of the body along an arbitrary smooth curve can be implemented by rotating the rotors are presented. For the optimal control problem, equations of subRiemannian geodesics on $SE(3)$ are obtained.

Borisov A. V., Kazakov A. O., Pivovarova E. N.
Regular and Chaotic Dynamics in the Rubber Model of a Chaplygin Top
Abstract
This paper is concerned with the rolling motion of a dynamically asymmetric unbalanced ball (Chaplygin top) in a gravitational field on a plane under the assumption that there is no slipping and spinning at the point of contact. We give a description of strange attractors existing in the system and discuss in detail the scenario of how one of them arises via a sequence of perioddoubling bifurcations. In addition, we analyze the dynamics of the system in absolute space and show that in the presence of strange attractors in the system the behavior of the point of contact considerably depends on the characteristics of the attractor and can be both chaotic and nearly quasiperiodic.

Quillen A. C., Askari H., Kelley D. H., Friedmann T., Oakes P. W.
A Coin Vibrational Motor Swimming at Low Reynolds Number
Abstract
Lowcost coin vibrational motors, used in haptic feedback, exhibit rotational internal motion inside a rigid case. Because the motor case motion exhibits rotational symmetry, when placed into a fluid such as glycerin, the motor does not swim even though its vibrations induce steady streaming in the fluid. However, a piece of rubber foam stuck to the curved case and giving the motor neutral buoyancy also breaks the rotational symmetry allowing it to swim. We measured a 1 cm diameter coin vibrational motor swimming in glycerin at a speed of a body length in 3 seconds or at 3 mm/s. The swim speed puts the vibrational motor in a low Reynolds number regime similar to bacterial motility, but because of the vibration it is not analogous to biological organisms. Rather the swimming vibrational motor may inspire small inexpensive robotic swimmers that are robust as they contain no external moving parts.
A time dependent Stokes equation planar sheet model suggests that the swim speed depends on a steady streaming velocity $V_{stream} \sim Re_s^{1/2} U_0$ where $U_0$ is the velocity of surface vibrations, and streaming Reynolds number $Re_s = U_0^2/(\omega \nu)$ for angular vibrational frequency $\omega$ and fluid
kinematic viscosity $\nu$.

Karavaev Y. L., Kilin A. A., Klekovkin A. V.
Experimental Investigations of the Controlled Motion of a Screwless Underwater Robot
Abstract
In this paper we describe the results of experimental investigations of the motion of a screwless underwater robot controlled by rotating internal rotors. We present the results of comparison of the trajectories obtained with the results of numerical simulation using the model of an ideal fluid.

Klenov A. I., Kilin A. A.
Influence of Vortex Structures on the Controlled Motion of an Abovewater Screwless Robot
Abstract
This paper is devoted to an experimental investigation of the motion of a rigid body set in motion by rotating two unbalanced internal masses. The results of experiments confirming the possibility of motion by this method are presented. The dependence of the parameters of motion on the rotational velocity of internal masses is analyzed. The velocity field of the fluid around the moving body is examined.

Borisov A. V., Kazakov A. O., Sataev I. R.
Spiral Chaos in the Nonholonomic Model of a Chaplygin Top
Abstract
This paper presents a numerical study of the chaotic dynamics of a dynamically asymmetric unbalanced ball (Chaplygin top) rolling on a plane. It is well known that the dynamics of such a system reduces to the investigation of a threedimensional map, which in the general case has no smooth invariant measure. It is shown that homoclinic strange attractors of discrete spiral type (discrete Shilnikov type attractors) arise in this model for certain parameters. From the viewpoint of physical motions, the trace of the contact point of a Chaplygin top on a plane is studied for the case where the phase trajectory sweeps out a discrete spiral attractor. Using the analysis of the trajectory of this trace, a conclusion is drawn
about the influence of “strangeness” of the attractor on the motion pattern of the top.
