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# Volume 22, Number 8, 2017

 Vetchanin  E. V.,  Mamaev I. S. Dynamics of Two Point Vortices in an External Compressible Shear Flow Abstract This paper is concerned with a system of equations that describes the motion of two point vortices in a flow possessing constant uniform vorticity and perturbed by an acoustic wave. The system is shown to have both regular and chaotic regimes of motion. In addition, simple and chaotic attractors are found in the system. Attention is given to bifurcations of fixed points of a Poincar´e map which lead to the appearance of these regimes. It is shown that, in the case where the total vortex strength changes, the “reversible pitch-fork” bifurcation is a typical scenario of emergence of asymptotically stable fixed and periodic points. As a result of this bifurcation, a saddle point, a stable and an unstable point of the same period emerge from an elliptic point of some period. By constructing and analyzing charts of dynamical regimes and bifurcation diagrams we show that a cascade of period-doubling bifurcations is a typical scenario of transition to chaos in the system under consideration. Keywords: point vortices, shear flow, perturbation by an acoustic wave, bifurcations, reversible pitch-fork, period doubling Citation: Vetchanin  E. V.,  Mamaev I. S., Dynamics of Two Point Vortices in an External Compressible Shear Flow, Regular and Chaotic Dynamics, 2017, vol. 22, no. 8, pp. 893–908 DOI:10.1134/S1560354717080019
 Ardentov A. A.,  Sachkov Y. L. Maxwell Strata and Cut Locus in the Sub-Riemannian Problem on the Engel Group Abstract We consider the nilpotent left-invariant sub-Riemannian structure on the Engel group. This structure gives a fundamental local approximation of a generic rank 2 sub-Riemannian structure on a 4-manifold near a generic point (in particular, of the kinematic models of a car with a trailer). On the other hand, this is the simplest sub-Riemannian structure of step three. We describe the global structure of the cut locus (the set of points where geodesics lose their global optimality), the Maxwell set (the set of points that admit more than one minimizer), and the intersection of the cut locus with the caustic (the set of conjugate points along all geodesics). The group of symmetries of the cut locus is described: it is generated by a one-parameter group of dilations $\mathbb{R}_+$ and a discrete group of reflections $\mathbb{Z}_2 \times \mathbb{Z}_2 \times \mathbb{Z}_2$. The cut locus admits a stratification with 6 three-dimensional strata, 12 two-dimensional strata, and 2 one-dimensional strata. Three-dimensional strata of the cut locus are Maxwell strata of multiplicity 2 (for each point there are 2 minimizers). Two-dimensional strata of the cut locus consist of conjugate points. Finally, one-dimensional strata are Maxwell strata of infinite multiplicity, they consist of conjugate points as well. Projections of sub-Riemannian geodesics to the 2-dimensional plane of the distribution are Euler elasticae. For each point of the cut locus, we describe the Euler elasticae corresponding to minimizers coming to this point. Finally, we describe the structure of the optimal synthesis, i. e., the set of minimizers for each terminal point in the Engel group. Keywords: sub-Riemannian geometry, optimal control, Engel group, Maxwell strata, cut locus, mobile robot, Euler's elasticae Citation: Ardentov A. A.,  Sachkov Y. L., Maxwell Strata and Cut Locus in the Sub-Riemannian Problem on the Engel Group, Regular and Chaotic Dynamics, 2017, vol. 22, no. 8, pp. 909–936 DOI:10.1134/S1560354717080020
 Chernyshev V. L.,  Tolchennikov A. A. The Second Term in the Asymptotics for the Number of Points Moving Along a Metric Graph Abstract We consider the problem of determining the asymptotics for the number of points moving along a metric graph. This problem is motivated by the problem of the evolution of wave packets, which at the initial moment of time are localized in a small neighborhood of one point. It turns out that the number of points, as a function of time, allows a polynomial approximation. This polynomial is expressed via Barnes’ multiple Bernoulli polynomials, which are related to the problem of counting the number of lattice points in expanding simplexes. In this paper we give explicit formulas for the first two terms of the expansion for the counting function of the number of moving points. The leading term was found earlier and depends only on the number of vertices, the number of edges and the lengths of the edges. The second term in the expansion shows what happens to the graph when one or two edges are removed. In particular, whether it breaks up into several connected components or not. In this paper, examples of the calculation of the leading and second terms are given. Keywords: metric graphs, Barnes’ multiple Bernoulli polynomials, lattice points, dynamical systems Citation: Chernyshev V. L.,  Tolchennikov A. A., The Second Term in the Asymptotics for the Number of Points Moving Along a Metric Graph, Regular and Chaotic Dynamics, 2017, vol. 22, no. 8, pp. 937–948 DOI:10.1134/S1560354717080032
 Mashtakov A. P.,  Popov A. Y. Extremal Controls in the Sub-Riemannian Problem on the Group of Motions of Euclidean Space Abstract For the sub-Riemannian problem on the group of motions of Euclidean space we present explicit formulas for extremal controls in the special case where one of the initial momenta is fixed. Keywords: sub-Riemannian geometry, special Euclidean motion group, extremal controls Citation: Mashtakov A. P.,  Popov A. Y., Extremal Controls in the Sub-Riemannian Problem on the Group of Motions of Euclidean Space, Regular and Chaotic Dynamics, 2017, vol. 22, no. 8, pp. 949–954 DOI:10.1134/S1560354717080044
 Bizyaev I. A.,  Borisov A. V.,  Mamaev I. S. The Chaplygin Sleigh with Parametric Excitation: Chaotic Dynamics and Nonholonomic Acceleration Abstract This paper is concerned with the Chaplygin sleigh with time-varying mass distribution (parametric excitation). The focus is on the case where excitation is induced by a material point that executes periodic oscillations in a direction transverse to the plane of the knife edge of the sleigh. In this case, the problem reduces to investigating a reduced system of two first-order equations with periodic coefficients, which is similar to various nonlinear parametric oscillators. Depending on the parameters in the reduced system, one can observe different types of motion, including those accompanied by strange attractors leading to a chaotic (diffusion) trajectory of the sleigh on the plane. The problem of unbounded acceleration (an analog of Fermi acceleration) of the sleigh is examined in detail. It is shown that such an acceleration arises due to the position of the moving point relative to the line of action of the nonholonomic constraint and the center of mass of the platform. Various special cases of existence of tensor invariants are found. Keywords: nonholonomic mechanics, Fermi acceleration, Chaplygin sleigh, parametric oscillator, tensor invariants, involution, strange attractor, Lyapunov exponents, reversible systems, chaotic dynamics Citation: Bizyaev I. A.,  Borisov A. V.,  Mamaev I. S., The Chaplygin Sleigh with Parametric Excitation: Chaotic Dynamics and Nonholonomic Acceleration, Regular and Chaotic Dynamics, 2017, vol. 22, no. 8, pp. 955–975 DOI:10.1134/S1560354717080056
 Sokolov S. V.,  Ryabov P. E. Bifurcation Analysis of the Dynamics of Two Vortices in a Bose – Einstein Condensate. The Case of Intensities of Opposite Signs Abstract This paper is concerned with a system two point vortices in a Bose – Einstein condensate enclosed in a trap. The Hamiltonian form of equations of motion is presented and its Liouville integrability is shown. A bifurcation diagram is constructed, analysis of bifurcations of Liouville tori is carried out for the case of opposite-signed vortices, and the types of critical motions are identified. Keywords: integrable Hamiltonian systems, Bose – Einstein condensate, point vortices, bifurcation analysis Citation: Sokolov S. V.,  Ryabov P. E., Bifurcation Analysis of the Dynamics of Two Vortices in a Bose – Einstein Condensate. The Case of Intensities of Opposite Signs, Regular and Chaotic Dynamics, 2017, vol. 22, no. 8, pp. 976–995 DOI:10.1134/S1560354717080068

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