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# Volume 16, Number 6, 2011

 O'Neil K. A. Clustered Equilibria of Point Vortices Abstract Point vortex equilibria in which the vortices are arranged in clusters are examined. The vortex velocities in these configurations are all equal. Necessary conditions for their existence are established that relate the circulations within the clusters to the cluster radius. A method for generating these configurations by singular continuation is proved to be valid for the generic case. Finally, a partial analysis of exceptional cases is given and their connection to the existence of parametrized families of equilibria is described. Keywords: point vortex, equilibrium, singular continuation Citation: O'Neil K. A., Clustered Equilibria of Point Vortices, Regular and Chaotic Dynamics, 2011, vol. 16, no. 6, pp. 555-561 DOI:10.1134/S1560354711060013
 Demina M. V.,  Kudryashov N. A. Point Vortices and Polynomials of the Sawada–Kotera and Kaup–Kupershmidt Equations Abstract Rational solutions and special polynomials associated with the generalized $K_2$ hierarchy are studied. This hierarchy is related to the Sawada–Kotera and Kaup–Kupershmidt equations and some other integrable partial differential equations including the Fordy–Gibbons equation. Differential–difference relations and differential equations satisfied by the polynomials are derived. The relationship between these special polynomials and stationary configurations of point vortices with circulations $\Gamma$ and $−2\Gamma$ is established. Properties of the polynomials are studied. Differential–difference relations enabling one to construct these polynomials explicitly are derived. Algebraic relations satisfied by the roots of the polynomials are found. Keywords: point vortices, special polynomials, generalized $K_2$ hierarchy, Sawada–Kotera equation, Kaup–Kupershmidt equation, Fordy–Gibbons equation Citation: Demina M. V.,  Kudryashov N. A., Point Vortices and Polynomials of the Sawada–Kotera and Kaup–Kupershmidt Equations, Regular and Chaotic Dynamics, 2011, vol. 16, no. 6, pp. 562-576 DOI:10.1134/S1560354711060025
 Duarte R.,  Carton X.,  Capet X.,  Cherubin L. Trapped Instability and Vortex Formation by an Unstable Coastal Current Abstract This paper addresses the instability of a two-layer coastal current in a quasigeostrophic model; the potential vorticity (PV) structure of this current consists in two uniform cores, located at different depths, with finite width and horizontally shifted. This shift allows both barotropic and baroclinic instability for this current. The PV cores can be like-signed or opposite-signed, leading to their vertical alignment or to their hetonic coupling. These two aspects are novel compared to previous studies. For narrow vorticity cores, short waves dominate, associated with barotropic instability; for wider cores, longer waves are more unstable and are associated with baroclinic processes. Numerical experiments were performed on the $f$−plane with a finite-difference model. When both cores have like-signed PV, trapped instability develops during the nonlinear evolution: vertical alignment of the structures is observed. For narrow cores, short wave breaking occurs close to the coast; for wider cores, substantial turbulence results from the entrainment of ambient fluid into the coastal jet. When the two cores have opposite-signed PV, the nonlinear regimes range from short wave breaking to the ejection of dipoles or tripoles, via a regime of dipole oscillation near the wall. The Fourier analysis of the perturbed flow is appropriate to distinguish the regimes of short wave breaking, of dipole formation, and of turbulence, but not the differences between regimes involving only vortex pairs. To explain more precisely the regimes where two vortices (and their wall images) interact, a point vortex model is appropriate. Keywords: stability and instability of geophysical and astrophysical flows, vortex flows, rotating fluids, stability problems, applications to physics Citation: Duarte R.,  Carton X.,  Capet X.,  Cherubin L., Trapped Instability and Vortex Formation by an Unstable Coastal Current, Regular and Chaotic Dynamics, 2011, vol. 16, no. 6, pp. 577-601 DOI:10.1134/S1560354711060037
 Kozlov V. V. The Vlasov Kinetic Equation, Dynamics of Continuum and Turbulence Abstract We consider a continuum of interacting particles whose evolution is governed by the Vlasov kinetic equation. An infinite sequence of equations of motion for this medium (in the Eulerian description) is derived and its general properties are explored. An important example is a collisionless gas, which exhibits irreversible behavior. Though individual particles interact via a potential, the dynamics of the continuum bears dissipative features. Applicability of the Vlasov equations to the modeling of small-scale turbulence is discussed. Keywords: kinetic Vlasov’s equation, Euler’s equation, continuum, turbulence Citation: Kozlov V. V., The Vlasov Kinetic Equation, Dynamics of Continuum and Turbulence, Regular and Chaotic Dynamics, 2011, vol. 16, no. 6, pp. 602-622 DOI:10.1134/S1560354711060049
 Ehlers K. M.,  Koiller J. Micro-swimming Without Flagella: Propulsion by Internal Structures Abstract Since a first proof-of-concept for an autonomous micro-swimming device appeared in 2005 a strong interest on the subject ensued. The most common configuration consists of a cell driven by an external propeller, bio-inspired by bacteria such as E.coli. It is natural to investigate whether micro-robots powered by internal mechanisms could be competitive. We compute the translational and rotational velocity of a spheroid that produces a helical wave on its surface, as has been suggested for the rod-shaped cyanobacterium Synechococcus. This organisms swims up to ten body lengths per second without external flagella. For the mathematical analysis we employ the tangent plane approximation method, which is adequate for amplitudes, frequencies and wave lengths considered here. We also present a qualitative discussion about the efficiency of a device driven by an internal rotating structure. Keywords: bio-inspired micro-swimming devices, Stokes flows, efficiency, Synechococcus Citation: Ehlers K. M.,  Koiller J., Micro-swimming Without Flagella: Propulsion by Internal Structures, Regular and Chaotic Dynamics, 2011, vol. 16, no. 6, pp. 623-652 DOI:10.1134/S1560354711060050
 Borisov A. V.,  Kilin A. A.,  Mamaev I. S. On the Model of Non-holonomic Billiard Abstract In this paper we develop a new model of non-holonomic billiard that accounts for the intrinsic rotation of the billiard ball. This model is a limit case of the problem of rolling without slipping of a ball without slipping over a quadric surface. The billiards between two parallel walls and inside a circle are studied in detail. Using the three-dimensional-point-map technique, the non-integrability of the non-holonomic billiard within an ellipse is shown. Keywords: billiard, impact, point map, nonintegrability, periodic solution, nonholonomic constraint, integral of motion Citation: Borisov A. V.,  Kilin A. A.,  Mamaev I. S., On the Model of Non-holonomic Billiard, Regular and Chaotic Dynamics, 2011, vol. 16, no. 6, pp. 653-662 DOI:10.1134/S1560354711060062
 Kim B. Routh Symmetry in the Chaplygin’s Rolling Ball Abstract The Routh integral in the symmetric Chaplygin’s rolling ball has been regarded as a mysterious conservation law due to its interesting form of $\sqrt{I_1I_3 + m ⟨Is, s⟩}\Omega_3$. In this paper, a new form of the Routh integral is proposed as a Noether’s pairing form of a conservation law. An explicit symmetry vector for the Routh integral is proved to associate the conserved quantity with the invariance of the Lagrangian function under the rollingly constrained nonholonomic variation. Then, the form of the Routh symmetry vector is discussed for its origin as the linear combination of the configurational vectors. Keywords: non-holonomic system, Noether symmetry, integrable system, Lagrange–D’Alembert equations Citation: Kim B., Routh Symmetry in the Chaplygin’s Rolling Ball, Regular and Chaotic Dynamics, 2011, vol. 16, no. 6, pp. 663-670 DOI:10.1134/S1560354711060074
 Borisov A. V.,  Meleshko V. V.,  Stremler M. A.,  van Heijst G. J. Hassan Aref (1950–2011) Abstract Citation: Borisov A. V.,  Meleshko V. V.,  Stremler M. A.,  van Heijst G. J., Hassan Aref (1950–2011), Regular and Chaotic Dynamics, 2011, vol. 16, no. 6, pp. 671-684 DOI:10.1134/S1560354711060086

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