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2013
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# Volume 18, Number 4, 2013

 Kozlov V. V. The Euler–Jacobi–Lie Integrability Theorem Abstract This paper addresses a class of problems associated with the conditions for exact integrability of systems of ordinary differential equations expressed in terms of the properties of tensor invariants. The general theorem of integrability of the system of $n$ differential equations is proved, which admits $n−2$ independent symmetry fields and an invariant volume $n$-form (integral invariant). General results are applied to the study of steady motions of a continuum with infinite conductivity. Keywords: symmetry field, integral invariant, nilpotent group, magnetic hydrodynamics Citation: Kozlov V. V., The Euler–Jacobi–Lie Integrability Theorem, Regular and Chaotic Dynamics, 2013, vol. 18, no. 4, pp. 329-343 DOI:10.1134/S1560354713040011
 Demina M. V.,  Kudryashov N. A. Relative Equilibrium Configurations of Point Vortices on a Sphere Abstract The problem of constructing and classifying equilibrium and relative equilibrium configurations of point vortices on a sphere is studied. A method which enables one to find any such configuration is presented. Configurations formed by the vortices placed at the vertices of Platonic solids are considered without making the assumption that the vortices possess equal in absolute value circulations. Several new configurations are obtained. Keywords: point vortices, sphere, relative equilibrium, fixed equilibrium, Platonic solids Citation: Demina M. V.,  Kudryashov N. A., Relative Equilibrium Configurations of Point Vortices on a Sphere, Regular and Chaotic Dynamics, 2013, vol. 18, no. 4, pp. 344-355 DOI:10.1134/S1560354713040023
 Borisov A. V.,  Mamaev I. S. Topological Analysis of an Integrable System Related to the Rolling of a Ball on a Sphere Abstract A new integrable system describing the rolling of a rigid body with a spherical cavity on a spherical base is considered. Previously the authors found the separation of variables for this system on the zero level set of a linear (in angular velocity) first integral, whereas in the general case it is not possible to separate the variables. In this paper we show that the foliation into invariant tori in this problem is equivalent to the corresponding foliation in the Clebsch integrable system in rigid body dynamics (for which no real separation of variables has been found either). In particular, a fixed point of focus type is possible for this system, which can serve as a topological obstacle to the real separation of variables. Keywords: integrable system, bifurcation diagram, conformally Hamiltonian system, bifurcation, Liouville foliation, critical periodic solution Citation: Borisov A. V.,  Mamaev I. S., Topological Analysis of an Integrable System Related to the Rolling of a Ball on a Sphere, Regular and Chaotic Dynamics, 2013, vol. 18, no. 4, pp. 356-371 DOI:10.1134/S1560354713040035
 García-Naranjo L. C.,  Marrero J. C. Non-Existence of an Invariant Measure for a Homogeneous Ellipsoid Rolling on the Plane Abstract It is known that the reduced equations for an axially symmetric homogeneous ellipsoid that rolls without slipping on the plane possess a smooth invariant measure. We show that such an invariant measure does not exist in the case when all of the semi-axes of the ellipsoid have different length. Keywords: nonholonomic mechanical systems, invariant volume forms, symmetries, reduction Citation: García-Naranjo L. C.,  Marrero J. C., Non-Existence of an Invariant Measure for a Homogeneous Ellipsoid Rolling on the Plane, Regular and Chaotic Dynamics, 2013, vol. 18, no. 4, pp. 372-379 DOI:10.1134/S1560354713040047
 Jing F.,  Kanso E. Stability of Underwater Periodic Locomotion Abstract Most aquatic vertebrates swim by lateral flapping of their bodies and caudal fins. While much effort has been devoted to understanding the flapping kinematics and its influence on the swimming efficiency, little is known about the stability (or lack of) of periodic swimming. It is believed that stability limits maneuverability and body designs/flapping motions that are adapted for stable swimming are not suitable for high maneuverability and vice versa. In this paper, we consider a simplified model of a planar elliptic body undergoing prescribed periodic heaving and pitching in potential flow. We show that periodic locomotion can be achieved due to the resulting hydrodynamic forces, and its value depends on several parameters including the aspect ratio of the body, the amplitudes and phases of the prescribed flapping.We obtain closedform solutions for the locomotion and efficiency for small flapping amplitudes, and numerical results for finite flapping amplitudes. This efficiency analysis results in optimal parameter values that are in agreement with values reported for some carangiform fish. We then study the stability of the (finite amplitude flapping) periodic locomotion using Floquet theory. We find that stability depends nonlinearly on all parameters. Interesting trends of switching between stable and unstable motions emerge and evolve as we continuously vary the parameter values. This suggests that, for live organisms that control their flapping motion, maneuverability and stability need not be thought of as disjoint properties, rather the organism may manipulate its motion in favor of one or the other depending on the task at hand. Keywords: biolocomotion, solid-fluid interactions, efficiency, motion stability Citation: Jing F.,  Kanso E., Stability of Underwater Periodic Locomotion, Regular and Chaotic Dynamics, 2013, vol. 18, no. 4, pp. 380-393 DOI:10.1134/S1560354713040059
 Valent G. On a Class of Integrable Systems with a Quartic First Integral Abstract We generalize, to some extent, the results on integrable geodesic flows on two dimensional manifolds with a quartic first integral in the framework laid down by Selivanova and Hadeler. The local structure is first determined by a direct integration of the differential system which expresses the conservation of the quartic observable and is seen to involve a finite number of parameters. The global structure is studied in some detail and leads to a class of models on the manifolds $\mathbb{S}^2$, $\mathbb{H}^2$ or $\mathbb{R}^2$. As special cases we recover Kovalevskaya’s integrable system and a generalization of it due to Goryachev. Keywords: integrable Hamiltonian systems, quartic polynomial integral, manifolds for Riemannian metrics Citation: Valent G., On a Class of Integrable Systems with a Quartic First Integral, Regular and Chaotic Dynamics, 2013, vol. 18, no. 4, pp. 394-424 DOI:10.1134/S1560354713040060
 Del Magno G.,  Markarian R. Singular Sets of Planar Hyperbolic Billiards are Regular Abstract Many planar hyperbolic billiards are conjectured to be ergodic. This paper represents a first step towards the proof of this conjecture. The Hopf argument is a standard technique for proving the ergodicity of a smooth hyperbolic system. Under additional hypotheses, this technique also applies to certain hyperbolic systems with singularities, including hyperbolic billiards. The supplementary hypotheses concern the subset of the phase space where the system fails to be $C^2$ differentiable. In this work, we give a detailed proof of one of these hypotheses for a large collection of planar hyperbolic billiards. Namely, we prove that the singular set and each of its iterations consist of a finite number of compact curves of class $C^2$ with finitely many intersection points. Keywords: hyperbolic billiards, ergodicity Citation: Del Magno G.,  Markarian R., Singular Sets of Planar Hyperbolic Billiards are Regular, Regular and Chaotic Dynamics, 2013, vol. 18, no. 4, pp. 425-452 DOI:10.1134/S1560354713040072
 Rauch-Wojciechowski S.,  Rutstam N. Dynamics of the Tippe Top — Properties of Numerical Solutions Versus the Dynamical Equations Abstract We study the relationship between numerical solutions for inverting Tippe Top and the structure of the dynamical equations. The numerical solutions confirm the oscillatory behavior of the inclination angle $\theta(t)$ for the symmetry axis of the Tippe Top, as predicted by the Main Equation for the Tippe Top. They also reveal further fine features of the dynamics of inverting solutions defining the time of inversion. These features are partially understood on the basis of the underlying dynamical equations. Keywords: Tippe Top, rigid body, nonholonomic mechanics, numerical solutions Citation: Rauch-Wojciechowski S.,  Rutstam N., Dynamics of the Tippe Top — Properties of Numerical Solutions Versus the Dynamical Equations, Regular and Chaotic Dynamics, 2013, vol. 18, no. 4, pp. 453-467 DOI:10.1134/S1560354713040084

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