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# Volume 22, Number 5, 2017

 Carton X.,  Morvan M.,  Reinaud J. N.,  Sokolovskiy M. A.,  L’Hegaret P.,  Vic C. Vortex Merger near a Topographic Slope in a Homogeneous Rotating Fluid Abstract The effect of a bottom slope on the merger of two identical Rankine vortices is investigated in a two-dimensional, quasi-geostrophic, incompressible fluid. When two cyclones initially lie parallel to the slope, and more than two vortex diameters away from the slope, the critical merger distance is unchanged. When the cyclones are closer to the slope, they can merge at larger distances, but they lose more mass into filaments, thus weakening the efficiency of merger. Several effects account for this: the topographic Rossby wave advects the cyclones, reduces their mutual distance and deforms them. This alongshelf wave breaks into filaments and into secondary vortices which shear out the initial cyclones. The global motion of fluid towards the shallow domain and the erosion of the two cyclones are confirmed by the evolution of particles seeded both in the cyclones and near the topographic slope. The addition of tracer to the flow indicates that diffusion is ballistic at early times. For two anticyclones, merger is also facilitated because one vortex is ejected offshore towards the other, via coupling with a topographic cyclone. Again two anticyclones can merge at large distance but they are eroded in the process. Finally, for taller topographies, the critical merger distance is again increased and the topographic influence can scatter or completely erode one of the two initial cyclones. Conclusions are drawn on possible improvements of the model configuration for an application to the ocean. Keywords: two-dimensional incompressible flow, vortex merger, critical merger distance, bottom slope, topographic wave and vortices, diffusion Citation: Carton X.,  Morvan M.,  Reinaud J. N.,  Sokolovskiy M. A.,  L’Hegaret P.,  Vic C., Vortex Merger near a Topographic Slope in a Homogeneous Rotating Fluid, Regular and Chaotic Dynamics, 2017, vol. 22, no. 5, pp. 455-478 DOI:10.1134/S156035471705001X
 Ivanov A. V. Connecting Orbits near the Adiabatic Limit of Lagrangian Systems with Turning Points Abstract We consider a natural Lagrangian system defined on a complete Riemannian manifold being subjected to action of a time-periodic force field with potential $U(q,t, \varepsilon) = f(\varepsilon t)V(q)$ depending slowly on time. It is assumed that the factor $f(\tau)$ is periodic and vanishes at least at one point on the period. Let $X_{c}$ denote a set of isolated critical points of $V(x)$ at which $V(x)$ distinguishes its maximum or minimum. In the adiabatic limit $\varepsilon \to 0$ we prove the existence of a set $\mathcal{E}_{h}$ such that the system possesses a rich class of doubly asymptotic trajectories connecting points of $X_{c}$  for $\varepsilon \in \mathcal{E}_{h}$. Keywords: connecting orbits, homoclinic and heteroclinic orbits, nonautonomous Lagrangian system, singular perturbation, exponential dichotomy Citation: Ivanov A. V., Connecting Orbits near the Adiabatic Limit of Lagrangian Systems with Turning Points, Regular and Chaotic Dynamics, 2017, vol. 22, no. 5, pp. 479-501 DOI:10.1134/S1560354717050021
 Kruglikov B.,  Vollmer A.,  Lukes-Gerakopoulols G. On Integrability of Certain Rank 2 Sub-Riemannian Structures Abstract We discuss rank 2 sub-Riemannian structures on low-dimensional manifolds and prove that some of these structures in dimensions 6, 7 and 8 have a maximal amount of symmetry but no integrals polynomial in momenta of low degrees, except for those coming from the Killing vector fields and the Hamiltonian, thus indicating nonintegrability of the corresponding geodesic flows. Keywords: Sub-Riemannian geodesic flow, Killing tensor, integral, symmetry, Tanaka prolongation, overdetermined system of PDE, prolongation Citation: Kruglikov B.,  Vollmer A.,  Lukes-Gerakopoulols G., On Integrability of Certain Rank 2 Sub-Riemannian Structures, Regular and Chaotic Dynamics, 2017, vol. 22, no. 5, pp. 502-519 DOI:10.1134/S1560354717050033
 Gonzalez Leon M. A.,  Mateos Guilarte J.,  de la Torre Mayado M. Orbits in the Problem of Two Fixed Centers on the Sphere Abstract A trajectory isomorphism between the two Newtonian fixed center problem in the sphere and two associated planar two fixed center problems is constructed by performing two simultaneous gnomonic projections in $S^2$. This isomorphism converts the original quadratures into elliptic integrals and allows the bifurcation diagram of the spherical problem to be analyzed in terms of the corresponding ones of the planar systems. The dynamics along the orbits in the different regimes for the problem in $S^2$ is expressed in terms of Jacobi elliptic functions. Keywords: spherical two-center problem, separation of variables, spheroconical coordinates, elliptic coordinates Citation: Gonzalez Leon M. A.,  Mateos Guilarte J.,  de la Torre Mayado M., Orbits in the Problem of Two Fixed Centers on the Sphere, Regular and Chaotic Dynamics, 2017, vol. 22, no. 5, pp. 520-542 DOI:10.1134/S1560354717050045
 Llibre J.,  Valls  C. Darboux Polynomials, Balances and Painlevé Property Abstract For a given polynomial differential system we provide different necessary conditions for the existence of Darboux polynomials using the balances of the system and the Painlevé property. As far as we know, these are the first results which relate the Darboux theory of integrability, first, to the Painlevé property and, second, to the Kovalevskaya exponents. The relation of these last two notions to the general integrability has been intensively studied over these last years. Keywords: Painlevé property, Darboux polynomial, Kovalevskaya exponents Citation: Llibre J.,  Valls  C., Darboux Polynomials, Balances and Painlevé Property, Regular and Chaotic Dynamics, 2017, vol. 22, no. 5, pp. 543-550 DOI:10.1134/S1560354717050057
 Kuptsov P. V.,  Kuznetsov S. P.,  Stankevich N. V. A Family of Models with Blue Sky Catastrophes of Different Classes Abstract A generalized model with bifurcations associated with blue sky catastrophes is introduced. Depending on an integer index $m$, different kinds of attractors arise, including those associated with quasi-periodic oscillations and with hyperbolic chaos. Verification of the hyperbolicity is provided based on statistical analysis of intersection angles of stable and unstable manifolds. Keywords: dynamical system, blue sky catastrophe, quasi-periodic oscillations, hyperbolic chaos, Smale–Williams solenoid Citation: Kuptsov P. V.,  Kuznetsov S. P.,  Stankevich N. V., A Family of Models with Blue Sky Catastrophes of Different Classes, Regular and Chaotic Dynamics, 2017, vol. 22, no. 5, pp. 551-565 DOI:10.1134/S1560354717050069
 Pepa R. Y.,  Popelensky T. Y. Equilibrium for a Combinatorial Ricci Flow with Generalized Weights on a Tetrahedron Abstract Chow and Lou [2] showed in 2003 that under certain conditions the combinatorial analogue of the Hamilton Ricci flow on surfaces converges to Thruston’s circle packing metric of constant curvature. The combinatorial setting includes weights defined for edges of a triangulation. A crucial assumption in [2] was that the weights are nonnegative.We have recently shown that the same statement on convergence can be proved under weaker conditions: some weights can be negative and should satisfy certain inequalities. In this note we show that there are some restrictions for weakening the conditions. Namely, we show that in some situations the combinatorial Ricci flow has no equilibrium or has several points of equilibrium and, in particular, the convergence theorem is no longer valid. Keywords: circle packing, combinatorial Ricci flow Citation: Pepa R. Y.,  Popelensky T. Y., Equilibrium for a Combinatorial Ricci Flow with Generalized Weights on a Tetrahedron, Regular and Chaotic Dynamics, 2017, vol. 22, no. 5, pp. 566-578 DOI:10.1134/S1560354717050070

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