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# Volume 23, Number 7-8, 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 two-dimensional 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. Keywords: Chaplygin sleigh, friction, parametric oscillator, strange attractor, Lyapunov exponents, chaotic dynamics, fish-like robot Citation: 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, Regular and Chaotic Dynamics, 2018, vol. 23, no. 7-8, pp. 803-820 DOI:10.1134/S1560354718070018
 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. Keywords: exponential stability, Nekhoroshev theory, perturbation theory, normal form theory, central force problem Citation: Bambusi D.,  Fusè A.,  Sansottera M., Exponential Stability in the Perturbed Central Force Problem, Regular and Chaotic Dynamics, 2018, vol. 23, no. 7-8, pp. 821-841 DOI:10.1134/S156035471807002X
 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. Keywords: semiclassical eigenvalues, complex vector bundles, holonomy group Citation: Shafarevich A. I., The Maslov Complex Germ and Semiclassical Spectral Series Corresponding to Singular Invariant Curves of Partially Integrable Hamiltonian Systems, Regular and Chaotic Dynamics, 2018, vol. 23, no. 7-8, pp. 842-849 DOI:10.1134/S1560354718070031
 Borisov A. V.,  Mamaev I. S.,  Vetchanin  E. V. Self-propulsion of a Smooth Body in a Viscous Fluid Under Periodic Oscillations of a Rotor and Circulation Abstract This paper addresses the problem of the self-propulsion 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 speed-up 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 period-doubling bifurcations. Keywords: self-propulsion in a fluid, smooth body, viscous fluid, periodic oscillation of circulation, control of a rotor Citation: Borisov A. V.,  Mamaev I. S.,  Vetchanin  E. V., Self-propulsion of a Smooth Body in a Viscous Fluid Under Periodic Oscillations of a Rotor and Circulation, Regular and Chaotic Dynamics, 2018, vol. 23, no. 7-8, pp. 850-874 DOI:10.1134/S1560354718070043
 Mamaev I. S.,  Vetchanin  E. V. The Self-propulsion 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 one-parameter bifurcation diagram are calculated. It is shown that strange attractors appear in the system due to a cascade of period-doubling bifurcations. Keywords: self-propulsion, Zhukovskii foil, foil with a sharp edge, motion in a viscous fluid, controlled motion, period-doubling bifurcation Citation: Mamaev I. S.,  Vetchanin  E. V., The Self-propulsion of a Foil with a Sharp Edge in a Viscous Fluid Under the Action of a Periodically Oscillating Rotor, Regular and Chaotic Dynamics, 2018, vol. 23, no. 7-8, pp. 875-886 DOI:10.1134/S1560354718070055
 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 right-hand 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. Keywords: integrable system, system with a discontinuous right-hand side, nonholonomic constraint, bifurcation diagram, body of revolution, sharp edge, wheel, rubber model Citation: Kilin A. A.,  Pivovarova E. N., Integrable Nonsmooth Nonholonomic Dynamics of a Rubber Wheel with Sharp Edges, Regular and Chaotic Dynamics, 2018, vol. 23, no. 7-8, pp. 887-907 DOI:10.1134/S1560354718070067
 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 well-known 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 finite-time 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 finite-time 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. Keywords: chaotic attractor, strange pseudohyperbolic attractor, method of angles, hyperbolic isolation, Lyapunov exponents, finite-time Lyapunov exponents, instant Lyapunov exponents, covariant Lyapunov vectors Citation: Kuptsov P. V.,  Kuznetsov S. P., Lyapunov Analysis of Strange Pseudohyperbolic Attractors: Angles Between Tangent Subspaces, Local Volume Expansion and Contraction, Regular and Chaotic Dynamics, 2018, vol. 23, no. 7-8, pp. 908-932 DOI:10.1134/S1560354718070079
 Yamanaka S. Local Integrability of Poincaré-Dulac Normal Forms Abstract We consider dynamical systems in Poincaré-Dulac 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 codimension-two fold-Hopf bifurcation. Keywords: Poincaré-Dulac normal form, integrability, dynamical system Citation: Yamanaka S., Local Integrability of Poincaré-Dulac Normal Forms, Regular and Chaotic Dynamics, 2018, vol. 23, no. 7-8, pp. 933-947 DOI:10.1134/S1560354718070080
 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 mean-square deviation of the structure in the ith layer on the nonlocal coupling parameters. Keywords: synchronization, many layer network, chimera states, nonlocal coupling, unidirectional coupling Citation: Strelkova G. I.,  Vadivasova T. E.,  Anishchenko V. S., Synchronization of Chimera States in a Network of Many Unidirectionally Coupled Layers of Discrete Maps, Regular and Chaotic Dynamics, 2018, vol. 23, no. 7-8, pp. 948-960 DOI:10.1134/S1560354718070092
 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 $(\alpha \geqslant 2)$. Keywords: instability, homographic solutions, central configurations, Jacobi –Maupertuis metric Citation: Jackman C.,  Melèndez J., On the Sectional Curvature Along Central Configurations, Regular and Chaotic Dynamics, 2018, vol. 23, no. 7-8, pp. 961-973 DOI:10.1134/S1560354718070109
 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 saddle-foci equilibria. Keywords: phase oscillators, heteroclinic networks Citation: Grines E. A.,  Osipov G. V., Heteroclinic and Homoclinic Structures in the System of Four Identical Globally Coupled Phase Oscillators with Nonpairwise Interactions, Regular and Chaotic Dynamics, 2018, vol. 23, no. 7-8, pp. 974-982 DOI:10.1134/S1560354718070110
 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. Keywords: Roller Racer, speed-up, nonholonomic mechanics, Rayleigh dissipation function, viscous friction, integrability by quadratures, control, constraint reaction force Citation: Bizyaev I. A.,  Borisov A. V.,  Mamaev I. S., Exotic Dynamics of Nonholonomic Roller Racer with Periodic Control, Regular and Chaotic Dynamics, 2018, vol. 23, no. 7-8, pp. 983-994 DOI:10.1134/S1560354718070122

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