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# Volume 20, Number 6, 2015

 Bardin B. S.,  Lanchares V. On the Stability of Periodic Hamiltonian Systems with One Degree of Freedom in the Case of Degeneracy Abstract We deal with the stability problem of an equilibrium position of a periodic Hamiltonian system with one degree of freedom. We suppose the Hamiltonian is analytic in a small neighborhood of the equilibrium position, and the characteristic exponents of the linearized system have zero real part, i.e., a nonlinear analysis is necessary to study the stability in the sense of Lyapunov. In general, the stability character of the equilibrium depends on nonzero terms of the lowest order $N (N >2)$ in the Hamiltonian normal form, and the stability problem can be solved by using known criteria. We study the so-called degenerate cases, when terms of order higher than $N$ must be taken into account to solve the stability problem. For such degenerate cases, we establish general conditions for stability and instability. Besides, we apply these results to obtain new stability criteria for the cases of degeneracy, which appear in the presence of first, second, third and fourth order resonances. Keywords: Hamiltonian systems, Lyapunov stability, stability theory, normal forms, KAM theory, Chetaev’s function, resonance Citation: Bardin B. S.,  Lanchares V., On the Stability of Periodic Hamiltonian Systems with One Degree of Freedom in the Case of Degeneracy, Regular and Chaotic Dynamics, 2015, vol. 20, no. 6, pp. 627-648 DOI:10.1134/S1560354715060015
 Kuznetsov S. P. Hyperbolic Chaos in Self-oscillating Systems Based on Mechanical Triple Linkage: Testing Absence of Tangencies of Stable and Unstable Manifolds for Phase Trajectories Abstract Dynamical equations are formulated and a numerical study is provided for selfoscillatory model systems based on the triple linkage hinge mechanism of Thurston – Weeks – Hunt – MacKay. We consider systems with a holonomic mechanical constraint of three rotators as well as systems, where three rotators interact by potential forces. We present and discuss some quantitative characteristics of the chaotic regimes (Lyapunov exponents, power spectrum). Chaotic dynamics of the models we consider are associated with hyperbolic attractors, at least, at relatively small supercriticality of the self-oscillating modes; that follows from numerical analysis of the distribution for angles of intersection of stable and unstable manifolds of phase trajectories on the attractors. In systems based on rotators with interacting potential the hyperbolicity is violated starting from a certain level of excitation. Keywords: dynamical system, chaos, hyperbolic attractor, Anosov dynamics, rotator, Lyapunov exponent, self-oscillator Citation: Kuznetsov S. P., Hyperbolic Chaos in Self-oscillating Systems Based on Mechanical Triple Linkage: Testing Absence of Tangencies of Stable and Unstable Manifolds for Phase Trajectories, Regular and Chaotic Dynamics, 2015, vol. 20, no. 6, pp. 649-666 DOI:10.1134/S1560354715060027
 Taimanov I. A. On an Integrable Magnetic Geodesic Flow on the Two-torus Abstract The magnetic geodesic flow on a flat two-torus with the magnetic field $F=\cos(x)dx \wedge dy$ is completely integrated and the description of all contractible periodic magnetic geodesics is given. It is shown that there are no such geodesics for energy $E \geqslant 1/2$, for $E<1/2$ simple periodic magnetic geodesics form two $S^1$-families for which the (fixed energy) action functional is positive and therefore there are no periodic magnetic geodesics for which the action functional is negative. Keywords: integrable system, magnetic geodesic flow Citation: Taimanov I. A., On an Integrable Magnetic Geodesic Flow on the Two-torus, Regular and Chaotic Dynamics, 2015, vol. 20, no. 6, pp. 667-678 DOI:10.1134/S1560354715060039
 Grines E. A.,  Osipov G. V. On Constructing Simple Examples of Three-dimensional Flows with Multiple Heteroclinic Cycles Abstract In this work we suggest a simple method for constructing $G$-equivariant systems of ODEs in $\mathbb{R}^3$ (i.e., systems whose trajectories are invariant under the action of this group on $\mathbb{R}^3$) that possess multiple disjoint heteroclinic networks. Heteroclinic networks under consideration consist of saddle equilibria that belong to coordinate axes and one-dimensional separatrices connecting them. We require these separatrices to lie on coordinate planes. We also assume the action of $G$ on $\mathbb{R}^3$ to be generated by cyclic permutation of coordinate variables and reflection with respect to one of the coordinate planes. As an example, we provide a step-by-step construction of three-dimensional flow with two disjoint heteroclinic networks. Also, we present a sketch of global dynamics analysis for the minimal example. Keywords: heteroclinic cycle, heteroclinic network Citation: Grines E. A.,  Osipov G. V., On Constructing Simple Examples of Three-dimensional Flows with Multiple Heteroclinic Cycles, Regular and Chaotic Dynamics, 2015, vol. 20, no. 6, pp. 679-690 DOI:10.1134/S1560354715060040
 Allilueva A. I.,  Shafarevich A. I. Asymptotic Solutions for Linear and Nonlinear MHD Systems with a Rapid Jump near a Surface. Dynamics of the Surface of the Jump and Evolution of the Magnetic Field Abstract We review our recent results concerning the asymptotic solutions for both linear and nonlinear MHD equations.We describe the asymptotic structure of the solution with a rapid jump near a 2D-surface. For the linear system we demonstrate the effect of the instantaneous growth of the solution. We also study the weak limit of the solution and the corresponding generalized problem. For the nonlinear system we describe the asymptotic division into different modes, the free boundary problem for the movement of the surface and the equation on the moving surface for the profile of the asymptotic solution. We also study the possibility of the instantaneous growth of the magnetic field. It appears that the growth is possible only in the case of the so-called degenerate Alfvén modes; the latter appear if the main term of the magnetic field is tangent to the surface of the jump. Keywords: MHD equations, discontinuous solutions, free boundary problems, dynamo theory, growth of the magnetic field Citation: Allilueva A. I.,  Shafarevich A. I., Asymptotic Solutions for Linear and Nonlinear MHD Systems with a Rapid Jump near a Surface. Dynamics of the Surface of the Jump and Evolution of the Magnetic Field, Regular and Chaotic Dynamics, 2015, vol. 20, no. 6, pp. 691-700 DOI:10.1134/S1560354715060052
 Korotkov A. G.,  Kazakov A. O.,  Osipov G. V. Sequential Dynamics in the Motif of Excitatory Coupled Elements Abstract In this article a new model of motif (small ensemble) of neuron-like elements is proposed. It is built with the use of the generalized Lotka–Volterra model with excitatory couplings. The main motivation for this work comes from the problems of neuroscience where excitatory couplings are proved to be the predominant type of interaction between neurons of the brain. In this paper it is shown that there are two modes depending on the type of coupling between the elements: the mode with a stable heteroclinic cycle and the mode with a stable limit cycle. Our second goal is to examine the chaotic dynamics of the generalized three-dimensional Lotka–Volterra model. Keywords: Neuronal motifs, Lotka–Volterra model, heteroclinic cycle, period-doubling bifurcation, Feigenbaum scenario, strange attractor, Lyapunov exponents Citation: Korotkov A. G.,  Kazakov A. O.,  Osipov G. V., Sequential Dynamics in the Motif of Excitatory Coupled Elements, Regular and Chaotic Dynamics, 2015, vol. 20, no. 6, pp. 701-715 DOI:10.1134/S1560354715060064
 Kilin A. A.,  Pivovarova E. N.,  Ivanova T. B. Spherical Robot of Combined Type: Dynamics and Control Abstract This paper is concerned with free and controlled motions of a spherical robot of combined type moving by displacing the center of mass and by changing the internal gyrostatic momentum. Equations of motion for the nonholonomic model are obtained and their first integrals are found. Fixed points of the reduced system are found in the absence of control actions. It is shown that they correspond to the motion of the spherical robot in a straight line and in a circle. A control algorithm for the motion of the spherical robot along an arbitrary trajectory is presented. A set of elementary maneuvers (gaits) is obtained which allow one to transfer the spherical robot from any initial point to any end point. Keywords: spherical robot, control, nonholonomic constraint, combined mechanism Citation: Kilin A. A.,  Pivovarova E. N.,  Ivanova T. B., Spherical Robot of Combined Type: Dynamics and Control, Regular and Chaotic Dynamics, 2015, vol. 20, no. 6, pp. 716-728 DOI:10.1134/S1560354715060076
 Novelia A.,  O'Reilly O. M. On Geodesics of the Rotation Group $SO(3)$ Abstract Geodesics on $SO(3)$ are characterized by constant angular velocity motions and as great circles on a three-sphere. The former interpretation is widely used in optometry and the latter features in the interpolation of rotations in computer graphics. The simplicity of these two disparate interpretations belies the complexity of the corresponding rotations. Using a quaternion representation for a rotation, we present a simple proof of the equivalence of the aforementioned characterizations and a straightforward method to establish features of the corresponding rotations. Keywords: quaternions, constraints, geodesics, Listing’s law, Slerp Citation: Novelia A.,  O'Reilly O. M., On Geodesics of the Rotation Group $SO(3)$, Regular and Chaotic Dynamics, 2015, vol. 20, no. 6, pp. 729-738 DOI:10.1134/S1560354715060088
 Borisov A. V.,  Mamaev I. S.,  Kilin A. A.,  Bizyaev I. A. Qualitative Analysis of the Dynamics of a Wheeled Vehicle Abstract This paper is concerned with the problem of the motion of a wheeled vehicle on a plane in the case where one of the wheel pairs is fixed. In addition, the motion of a wheeled vehicle on a plane in the case of two free wheel pairs is considered. A method for obtaining equations of motion for the vehicle with an arbitrary geometry is presented. Possible kinds of motion of the vehicle with a fixed wheel pair are determined. Keywords: nonholonomic constraint, system dynamics, wheeled vehicle, Chaplygin system Citation: Borisov A. V.,  Mamaev I. S.,  Kilin A. A.,  Bizyaev I. A., Qualitative Analysis of the Dynamics of a Wheeled Vehicle, Regular and Chaotic Dynamics, 2015, vol. 20, no. 6, pp. 739-751 DOI:10.1134/S156035471506009X
 Borisov A. V.,  Kilin A. A.,  Mamaev I. S. On the Hadamard – Hamel Problem and the Dynamics of Wheeled Vehicles Abstract In this paper, we develop the results obtained by J.Hadamard and G.Hamel concerning the possibility of substituting nonholonomic constraints into the Lagrangian of the system without changing the form of the equations of motion. We formulate the conditions for correctness of such a substitution for a particular case of nonholonomic systems in the simplest and universal form. These conditions are presented in terms of both generalized velocities and quasi-velocities. We also discuss the derivation and reduction of the equations of motion of an arbitrary wheeled vehicle. In particular, we prove the equivalence (up to additional quadratures) of problems of an arbitrary wheeled vehicle and an analogous vehicle whose wheels have been replaced with skates. As examples, we consider the problems of a one-wheeled vehicle and a wheeled vehicle with two rotating wheel pairs. Keywords: nonholonomic constraint, wheeled vehicle, reduction, equations of motion Citation: Borisov A. V.,  Kilin A. A.,  Mamaev I. S., On the Hadamard – Hamel Problem and the Dynamics of Wheeled Vehicles, Regular and Chaotic Dynamics, 2015, vol. 20, no. 6, pp. 752-766 DOI:10.1134/S1560354715060106

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