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# Volume 20, Number 2, 2015

 Silina K. G.,  Koroleva (Kikot) I. P.,  Manevitch L. I. Energy Exchange and Localization in the Planar Motion of a Weightless Beam Carrying Two Discrete Masses Abstract We present analytical and numerical studies of nonstationary resonance processes in a system with four degrees of freedom. The system under consideration can be considered as one of the simplest geometrically nonlinear discrete models of an elastic beam supported by nonlinear elastic grounding support. Two symmetrically distributed discrete masses reflect the inertial properties of the beam, two angular springs simulate its bending stiffness. The longitudinal springs, as is usual in systems of oscillators, reflect the tensile stiffness and two transversal springs simulate the reaction of grounding support. Dealing with lowenergy dynamics, we singled out the equations of transversal motion corresponding to the approximation of two coupled oscillators with nonlocal nonlinearity in elastic forces. We have analyzed this model using the concept of limiting phase trajectories (LPT). LPT’s concept was recently developed to study the nonstationary resonance dynamics. An analytical description of intensive interparticle energy exchange was obtained in terms of nonsmooth functions, which is consistent with numerical results. We have identified two dynamic transitions the first of which corresponds to the instability of out-of-phase normal mode and the second one is a transition from the intense energy exchange to the energy localization on the initially excited oscillator. Special attention was paid to the influence of bending stiffness on the conditions that ensure the implementation of each of the dynamic transitions. Keywords: energy exchange, energy localization, beam, elastic support, nonlinear normal modes, limiting phase trajectories Citation: Silina K. G.,  Koroleva (Kikot) I. P.,  Manevitch L. I., Energy Exchange and Localization in the Planar Motion of a Weightless Beam Carrying Two Discrete Masses, Regular and Chaotic Dynamics, 2015, vol. 20, no. 2, pp. 109-122 DOI:10.1134/S156035471502001X
 Kudryashov N. A. Analytical Solutions of the Lorenz System Abstract The Lorenz system is considered. The Painlevé test for the third-order equation corresponding to the Lorenz model at $\sigma \ne 0$ is presented. The integrable cases of the Lorenz system and the first integrals for the Lorenz system are discussed. The main result of the work is the classification of the elliptic solutions expressed via the Weierstrass function. It is shown that most of the elliptic solutions are degenerated and expressed via the trigonometric functions. However, two solutions of the Lorenz system can be expressed via the elliptic functions. Keywords: Lorenz system, Painlevé property, Painlevé test, analytical solutions, elliptic solutions Citation: Kudryashov N. A., Analytical Solutions of the Lorenz System, Regular and Chaotic Dynamics, 2015, vol. 20, no. 2, pp. 123-133 DOI:10.1134/S1560354715020021
 Karavaev Y. L.,  Kilin A. A. The Dynamics and Control of a Spherical Robot with an Internal Omniwheel Platform Abstract This paper deals with the problem of a spherical robot propelled by an internal omniwheel platform and rolling without slipping on a plane. The problem of control of spherical robot motion along an arbitrary trajectory is solved within the framework of a kinematic model and a dynamic model. A number of particular cases of motion are identified, and their stability is investigated. An algorithm for constructing elementary maneuvers (gaits) providing the transition from one steady-state motion to another is presented for the dynamic model. A number of experiments have been carried out confirming the adequacy of the proposed kinematic model. Keywords: spherical robot, kinematic model, dynamic model, nonholonomic constraint, omniwheel Citation: Karavaev Y. L.,  Kilin A. A., The Dynamics and Control of a Spherical Robot with an Internal Omniwheel Platform, Regular and Chaotic Dynamics, 2015, vol. 20, no. 2, pp. 134-152 DOI:10.1134/S1560354715020033
 Borisov A. V.,  Kilin A. A.,  Mamaev I. S. Dynamics and Control of an Omniwheel Vehicle Abstract A nonholonomic model of the dynamics of an omniwheel vehicle on a plane and a sphere is considered. A derivation of equations is presented and the dynamics of a free system are investigated. An explicit motion control algorithm for the omniwheel vehicle moving along an arbitrary trajectory is obtained. Keywords: omniwheel, roller-bearing wheel, nonholonomic constraint, dynamical system, invariant measure, integrability, controllability Citation: Borisov A. V.,  Kilin A. A.,  Mamaev I. S., Dynamics and Control of an Omniwheel Vehicle, Regular and Chaotic Dynamics, 2015, vol. 20, no. 2, pp. 153-172 DOI:10.1134/S1560354715020045
 Kashchenko A. A. Stability of Continuous Wave Solutions of One Laser Model with Large Delay Abstract Analysis of a delay differential laser model with large delay is presented. Sufficient conditions for existence of continuous wave solutions are found. It is shown that parameters determining the main part of asymptotics of these solutions lie on a bell-like curve. Sufficient conditions for stability of continuos wave solutions are found. The number of stability regions on bell-like curves is studied. It is proved that more than one region of stability may exist on these curves. It is shown that solutions with the same main part of asymptotics may have different stability properties if we change the value of linewidth enhancement factor. A mechanism for the destabilization of continuous wave solutions is found. Keywords: large delay, stability, laser dynamics, asymptotic methods Citation: Kashchenko A. A., Stability of Continuous Wave Solutions of One Laser Model with Large Delay, Regular and Chaotic Dynamics, 2015, vol. 20, no. 2, pp. 173-183 DOI:10.1134/S1560354715020057
 Shekutkovski N. One Property of Components of a Chain Recurrent Set Abstract For flows defined on a compact manifold with or without boundary, it is shown that the connectivity components of a chain recurrent set possess a stronger connectivity known as joinability (or pointed 1-movability in the sense of Borsuk). As a consequence, the Vietoris–van Dantzig solenoid cannot be a component of a chain recurrent set, although the solenoid appears as a minimal set of a flow. Keywords: chain recurrent set, continuity in a covering, pointed 1-movability, joinability Citation: Shekutkovski N., One Property of Components of a Chain Recurrent Set, Regular and Chaotic Dynamics, 2015, vol. 20, no. 2, pp. 184-188 DOI:10.1134/S1560354715020069
 Kuznetsov A. P.,  Migunova N. A.,  Sataev I. R.,  Sedova Y. V.,  Turukina L. V. From Chaos to Quasi-Periodicity Abstract Ensembles of several Rössler chaotic oscillators are considered. It is shown that a typical phenomenon for such systems is the emergence of different and sufficiently high dimensional invariant tori. The possibility of a quasi-periodic Hopf bifurcation and a cascade of such bifurcations based on tori of increasing dimension is demonstrated. The domains of resonance tori are revealed. Boundaries of these domains correspond to the saddle-node bifurcations. Inside the domains of resonance modes, torus-doubling bifurcations and destruction of tori are observed. Keywords: chaos, quasi-periodic oscillation, invariant torus, Lyapunov exponent, bifurcation Citation: Kuznetsov A. P.,  Migunova N. A.,  Sataev I. R.,  Sedova Y. V.,  Turukina L. V., From Chaos to Quasi-Periodicity, Regular and Chaotic Dynamics, 2015, vol. 20, no. 2, pp. 189-204 DOI:10.1134/S1560354715020070

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