Volume 23, Number 78
Volume 23, Number 78, 2018
Borisov A. V., Kuznetsov S. P.
Comparing Dynamics Initiated by an Attached Oscillating Particle for the Nonholonomic Model of a Chaplygin Sleigh and for a Model with Strong Transverse and Weak Longitudinal Viscous Friction Applied at a Fixed Point on the Body
Abstract
This paper addresses the problem of a rigid body moving on a plane (a platform) whose motion is initiated by oscillations of a point mass relative to the body in the presence of the viscous friction force applied at a fixed point of the platform and having in one direction a small (or even zero) value and a large value in the transverse direction. This problem is analogous to that of a Chaplygin sleigh when the nonholonomic constraint prohibiting motions of the fixed point on the platform across the direction prescribed on it is replaced by viscous friction. We present numerical results which confirm correspondence between the phenomenology of complex dynamics of the model with a nonholonomic constraint and a system with viscous friction —
phase portraits of attractors, bifurcation diagram, and Lyapunov exponents. In particular, we show the possibility of the platform’s motion being accelerated by oscillations of the internal mass, although, in contrast to the nonholonomic model, the effect of acceleration tends to saturation. We also show the possibility of chaotic dynamics related to strange attractors of equations for generalized velocities, which is accompanied by a twodimensional random walk of the platform in a laboratory reference system. The results obtained may be of interest to applications in the context of the problem of developing robotic mechanisms for motion in a fluid under the action of the motions of internal masses.

Bambusi D., Fusè A., Sansottera M.
Exponential Stability in the Perturbed Central Force Problem
Abstract
We consider the spatial central force problem with a real analytic potential. We prove that for all analytic potentials, but for the Keplerian and the harmonic ones, the Hamiltonian fulfills a nondegeneracy property needed for the applicability of Nekhoroshev’s
theorem. We deduce stability of the actions over exponentially long times when the system is subject to an arbitrary analytic perturbation. The case where the central system is put in interaction with a slow system is also studied and stability over exponentially long time is proved.

Shafarevich A. I.
The Maslov Complex Germ and Semiclassical Spectral Series Corresponding to Singular Invariant Curves of Partially Integrable Hamiltonian Systems
Abstract
We study semiclassical eigenvalues of the Schroedinger operator, corresponding to singular invariant curve of the corresponding classical system. The latter system is assumed to be partially integrable. We describe geometric object corresponding to the eigenvalues (comlex vector bundle over a graph) and compute semiclassical eigenvalues in terms of the corresponding holonomy group.

Borisov A. V., Mamaev I. S., Vetchanin E. V.
Selfpropulsion of a Smooth Body in a Viscous Fluid Under Periodic Oscillations of a Rotor and Circulation
Abstract
This paper addresses the problem of the selfpropulsion of a smooth body in a fluid by periodic oscillations of the internal rotor and circulation. In the case of zero dissipation and constant circulation, it is shown using methods of KAM theory that the kinetic energy of the system is a bounded function of time. In the case of constant nonzero circulation, the trajectories of the center of mass of the system lie in a bounded region of the plane. The method of expansion by a small parameter is used to approximately construct a solution corresponding to directed motion of a circular foil in the presence of dissipation and variable circulation.
Analysis of this approximate solution has shown that a speedup is possible in the system in the presence of variable circulation and in the absence of resistance to translational motion. It is shown that, in the case of an elliptic foil, directed motion is also possible. To explore the dynamics of the system in the general case, bifurcation diagrams, a chart of dynamical regimes
and a chart of the largest Lyapunov exponent are plotted. It is shown that the transition to chaos occurs through a cascade of perioddoubling bifurcations.

Mamaev I. S., Vetchanin E. V.
The Selfpropulsion of a Foil with a Sharp Edge in a Viscous Fluid Under the Action of a Periodically Oscillating Rotor
Abstract
This paper addresses the problem of controlled motion of the Zhukovskii foil in a viscous fluid due to a periodically oscillating rotor. Equations of motion including the added mass effect, viscous friction and lift force due to circulation are derived. It is shown that only limit cycles corresponding to the direct motion or motion near a circle appear in the system at the standard parameter values. The chart of dynamical regimes, the chart of the largest Lyapunov exponent and a oneparameter bifurcation diagram are calculated. It is shown that strange attractors appear in the system due to a cascade of perioddoubling bifurcations.

Kilin A. A., Pivovarova E. N.
Integrable Nonsmooth Nonholonomic Dynamics of a Rubber Wheel with Sharp Edges
Abstract
This paper is concerned with the dynamics of a wheel with sharp edges moving on a horizontal plane without slipping and rotation about the vertical (nonholonomic rubber model). The wheel is a body of revolution and has the form of a ball symmetrically truncated on both sides. This problem is described by a system of differential equations with a discontinuous
righthand side. It is shown that this system is integrable and reduces to quadratures. Partial solutions are found which correspond to fixed points of the reduced system. A bifurcation analysis and a classification of possible types of the wheel’s motion depending on the system parameters are presented.

Kuptsov P. V., Kuznetsov S. P.
Lyapunov Analysis of Strange Pseudohyperbolic Attractors: Angles Between Tangent Subspaces, Local Volume Expansion and Contraction
Abstract
Pseudohyperbolic attractors are genuine strange chaotic attractors. They do not contain stable periodic orbits and are robust in the sense that such orbits do not appear under variations. The tangent space of these attractors is split into a direct sum of volume expanding and contracting subspaces and these subspaces never have tangencies with each other. Any contraction in the first subspace, if it occurs, is weaker than contractions in the second one. In this paper we analyze the local structure of several chaotic attractors recently suggested in the literature as pseudohyperbolic. The absence of tangencies and thus the presence of the pseudohyperbolicity is verified using the method of angles that includes computation of distributions of the angles between the corresponding tangent subspaces. Also, we analyze how volume expansion in the first subspace and the contraction in the second one occurs locally. For this purpose we introduce a family of instant Lyapunov exponents. Unlike the wellknown
finite time ones, the instant Lyapunov exponents show expansion or contraction on infinitesimal time intervals. Two types of instant Lyapunov exponents are defined. One is related to ordinary finitetime Lyapunov exponents computed in the course of standard algorithm for Lyapunov exponents. Their sums reveal instant volume expanding properties. The second type of instant
Lyapunov exponents shows how covariant Lyapunov vectors grow or decay on infinitesimal time. Using both instant and finitetime Lyapunov exponents, we demonstrate that average expanding and contracting properties specific to pseudohyperbolicity are typically violated on infinitesimal time. Instantly volumes from the first subspace can sometimes be contacted, directions in the
second subspace can sometimes be expanded, and the instant contraction in the first subspace can sometimes be stronger than the contraction in the second subspace.

Yamanaka S.
Local Integrability of PoincaréDulac Normal Forms
Abstract
We consider dynamical systems in PoincaréDulac normal form having an equilibrium at the origin, and give a sufficient condition for them to be integrable, and prove that it is necessary for their special integrability under some condition. Moreover, we show that they are integrable if their resonance degrees are 0 or 1 and that they may be nonintegrable if their resonance degrees are greater than 1, as in Birkhoff normal forms for Hamiltonian systems. We demonstrate the theoretical results for a normal form appearing in the codimensiontwo foldHopf bifurcation.

Strelkova G. I., Vadivasova T. E., Anishchenko V. S.
Synchronization of Chimera States in a Network of Many Unidirectionally Coupled Layers of Discrete Maps
Abstract
We study numerically external synchronization of chimera states in a network of many unidirectionally coupled layers, each representing a ring of nonlocally coupled discretetime systems. The dynamics of each element in the network is described by either the logistic map or the bistable cubic map. We consider two cases: when all $M$ unidirectionally coupled layers are identical and when $(M  1)$ identical layers differ from the first driving layer in their nonlocal coupling parameters. It is shown that the master chimera state in the first layer can be retranslating along the network with small distortions which are defined by a parameter mismatch. We also analyze the dependence of the meansquare deviation of the structure in the ith layer on the nonlocal coupling parameters.

Jackman C., Melèndez J.
On the Sectional Curvature Along Central Configurations
Abstract
In this paper we characterize planar central configurations in terms of a sectional curvature value of the Jacobi–Maupertuis metric. This characterization works for the $N$body problem with general masses and any $1/r^\alpha$ potential with $\alpha > 0$. We also obtain dynamical consequences of these curvature values for relative equilibrium solutions. These curvature methods work well for strong forces $(\alpha \geqslant 2)$.

Grines E. A., Osipov G. V.
Heteroclinic and Homoclinic Structures in the System of Four Identical Globally Coupled Phase Oscillators with Nonpairwise Interactions
Abstract
Systems of $N$ identical globally coupled phase oscillators can demonstrate a multitude of complex behaviors. Such systems can have chaotic dynamics for $N > 4$ when a coupling function is biharmonic. The case $N = 4$ does not possess chaotic attractors when the coupling is biharmonic, but has them when the coupling includes nonpairwise interactions of phases. Previous studies have shown that some of chaotic attractors in this system are organized by heteroclinic networks. In the present paper we discuss which heteroclinic cycles are forbidden and which are supported by this particular system. We also discuss some of the cases regarding homoclinic trajectories to saddlefoci equilibria.

Bizyaev I. A., Borisov A. V., Mamaev I. S.
Exotic Dynamics of Nonholonomic Roller Racer with Periodic Control
Abstract
In this paper we consider the problem of the motion of the Roller Racer.We assume that the angle $\varphi (t)$ between the platforms is a prescribed function of time. We prove that in this case the acceleration of the Roller Racer is unbounded.
In this case, as the Roller Racer accelerates, the increase in the constraint reaction forces is also unbounded. Physically this means that, from a certain instant onward, the conditions of the rolling motion of the wheels without slipping are violated. Thus, we consider a model in which, in addition to the nonholonomic constraints, viscous friction force acts at the points of contact of the wheels. For this case we prove that there is no constant acceleration and all trajectories of the reduced system asymptotically tend to a periodic solution.
