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# Volume 26, Number 6, 2021 Special Issue: 200th birthday of Hermann von Helmholtz/Regular Papers

 Shashikanth B. N.,  Kidambi R. The Centroid-Deformation Decomposition for Buoyant Vortex Patch Motion Abstract The motion of a two-dimensional buoyant vortex patch, i. e., a vortex patch with a uniform density different from the uniform density of the surrounding fluid, is analyzed in terms of evolution equations for the motion of its centroid, deformation of its boundary and the strength distribution of a vortex sheet which is essential to enforce pressure continuity across the boundary. The equations for the centroid are derived by a linear momentum analysis and that for the sheet strength distribution by applying Euler’s equations on the boundary, while the boundary deformation is studied in the centroid-fixed frame. A complicated coupled set of equations is obtained which, to the best of our knowledge, has not been derived before. The evolution of the sheet strength distribution is obtained as an integral equation. The equations are also discussed in the limit of a patch of vanishing size or a buoyant point vortex. Keywords: deforming buoyant vortex patch, translating vortex centroid, vortex sheet, buoyant point vortex Citation: Shashikanth B. N.,  Kidambi R., The Centroid-Deformation Decomposition for Buoyant Vortex Patch Motion, Regular and Chaotic Dynamics, 2021, vol. 26, no. 6, pp. 577-599 DOI:10.1134/S1560354721060010
 Chu T.,  Llewellyn Smith S. G. Helical Contour Dynamics Abstract The equations of motion for an incompressible flow with helical symmetry (invariance under combined axial translation and rotation) can be expressed as nonlinear evolution laws for two scalars: vorticity and along-helix velocity. A metric term related to the pitch of the helix enters these equations, which reduce to two-dimensional and axisymmetric dynamics in appropriate limits. We take the vorticity and along-helix velocity component to be piecewise constant. In addition to this vortex patch, a vortex sheet develops when the along-helix velocity is nonzero.We obtain a contour dynamics formulation of the full nonlinear equations of motion, in which the motion of the boundary is computed in a Lagrangian fashion and the velocity field can be expressed as contour integrals, reducing the dimensionality of the computation. We investigate the stability properties of a circular vortex patch along the axis of the helix in the presence of a vortex sheet and along-helix velocity. A linear stability calculation shows that the system is stable when the initial vortex sheet is zero, but can be stable or unstable in the presence of a vortex sheet. Using contour dynamics, we examine the nonlinear evolution of the system, and show that nonlinear effects become important in unstable cases. Keywords: vortex dynamics, contour dynamics, vortex patch, vortex sheet, helical geometry Citation: Chu T.,  Llewellyn Smith S. G., Helical Contour Dynamics, Regular and Chaotic Dynamics, 2021, vol. 26, no. 6, pp. 600-617 DOI:10.1134/S1560354721060022
 Vic A.,  Carton X.,  Gula J. The Interaction of Two Unsteady Point Vortex Sources in a Deformation Field in 2D Incompressible Flows Abstract Taking into account the coupling of the ocean with the atmosphere is essential to properly describe vortex dynamics in the ocean. The forcing of a circular eddy with the relative wind stress curl leads to an Ekman pumping with a nonzero area integral. This in turn creates a source or a sink in the eddy. We revisit the two point vortex-source interaction, now coupled with an unsteady wind, leading to a time-varying circulation and source strength. Firstly, we recover the various fixed points of the two vortex-source system, and we calculate their stability. Then we show the effect of a weak amplitude, subharmonic, or harmonic time variation of the wind, leading to a similar variation of the circulation and the source strength of the vortex sources. We use a multiple time scale expansion of the variables to calculate the long time variation of these vortex trajectories around neutral fixed points. We study the amplitude equation and obtain its solution. We compute numerically the unstable evolution of the vortex sources when the source and circulation have a finite periodic variation. We also assess the influence of this time variation on the dispersion of a passive tracer near these vortex sources. Keywords: wind oceanic vortex interaction, vortex-source dynamics, system of ODE, equilibrium point and stability, multiple time scale expansion, dispersion of passive tracers Citation: Vic A.,  Carton X.,  Gula J., The Interaction of Two Unsteady Point Vortex Sources in a Deformation Field in 2D Incompressible Flows, Regular and Chaotic Dynamics, 2021, vol. 26, no. 6, pp. 618-646 DOI:10.1134/S1560354721060034
 Kozlov V. V. Integrals of Circulatory Systems Which are Quadratic in Momenta Abstract This paper addresses the problem of conditions for the existence of conservation laws (first integrals) of circulatory systems which are quadratic in velocities (momenta), when the external forces are nonpotential. Under some conditions the equations of motion are reduced to Hamiltonian form with some symplectic structure and the role of the Hamiltonian is played by a quadratic integral. In some cases the equations are reduced to a conformally Hamiltonian rather than Hamiltonian form. The existence of a quadratic integral and its properties allow conclusions to be drawn on the stability of equilibrium positions of circulatory systems. Keywords: circulatory system, polynomial integrals, Hamiltonian system, property of being conformally Hamiltonian, indices of inertia, asymptotic trajectories, Ziegler’s pendulum Citation: Kozlov V. V., Integrals of Circulatory Systems Which are Quadratic in Momenta, Regular and Chaotic Dynamics, 2021, vol. 26, no. 6, pp. 647-657 DOI:10.1134/S1560354721060046
 Bizyaev I. A.,  Mamaev I. S. Qualitative Analysis of the Dynamics of a Balanced Circular Foil and a Vortex Abstract This paper investigates the dynamics of a point vortex and a balanced circular foil in an ideal fluid. An explicit reduction to quadratures is performed. A bifurcation diagram is constructed and a classification of the types of integral manifolds is carried out. The stability of critical solutions is studied in which the foil and the vortex move in a circle or in a straight line. Keywords: point vortex, ideal fluid, bifurcation diagram, stability, relative equilibria Citation: Bizyaev I. A.,  Mamaev I. S., Qualitative Analysis of the Dynamics of a Balanced Circular Foil and a Vortex, Regular and Chaotic Dynamics, 2021, vol. 26, no. 6, pp. 658-674 DOI:10.1134/S1560354721060058
 Ramodanov S. M.,  Sokolov S. V. Dynamics of a Circular Cylinder and Two Point Vortices in a Perfect Fluid Abstract We study a mechanical system that consists of a 2D rigid body interacting dynamically with two point vortices in an unbounded volume of an incompressible, otherwise vortex-free, perfect fluid. The system has four degrees of freedom. The governing equations can be written in Hamiltonian form, are invariant under the action of the group $E$(2) and thus, in addition to the Hamiltonian function, admit three integrals of motion. Under certain restrictions imposed on the system’s parameters these integrals are in involution, thus rendering the system integrable (its order can be reduced by three degrees of freedom) and allowing for an analytical analysis of the dynamics. Keywords: point vortices, Hamiltonian systems, reduction Citation: Ramodanov S. M.,  Sokolov S. V., Dynamics of a Circular Cylinder and Two Point Vortices in a Perfect Fluid, Regular and Chaotic Dynamics, 2021, vol. 26, no. 6, pp. 675-691 DOI:10.1134/S156035472106006X
 Bogoyavlenskij O. I.,  Peng Y. Exact Solutions to the Beltrami Equation with a Non-constant $\alpha({\mathbf x})$ Abstract Infinite families of new exact solutions to the Beltrami equation with a non-constant $\alpha({\mathbf x})$ are derived. Differential operators connecting the steady axisymmetric Klein – Gordon equation and a special case of the Grad – Shafranov equation are constructed. A Lie semi-group of nonlinear transformations of the Grad – Shafranov equation is found. Keywords: ideal fluid equilibria, force-free plasma equilibria, Klein – Gordon equation, Yukawa potential, Beltrami equation Citation: Bogoyavlenskij O. I.,  Peng Y., Exact Solutions to the Beltrami Equation with a Non-constant $\alpha({\mathbf x})$, Regular and Chaotic Dynamics, 2021, vol. 26, no. 6, pp. 692-699 DOI:10.1134/S1560354721060071
 Petković B. Classification of Perturbations of Diophantine $\mathbb{Z}^m$ Actions on Tori of Arbitrary Dimension Abstract We generalize results of Moser  on the circle to $\mathbb{T}^d$: we show that a smooth sufficiently small perturbation of a $\mathbb Z^m$ action, $m \geqslant 2$, on the torus $\mathbb{T}^d$ by simultaneously Diophantine translations, is smoothly conjugate to the unperturbed action under a natural condition on the rotation sets of diffeomorphisms isotopic to identity and we answer the question Moser posed in  by proving the existence of a continuum of $m$-tuples of simultaneously Diophantine vectors such that every element of the induced $\mathbb Z^m$ action is Liouville. Keywords: KAM theory, simultaneously Diophantine translations, local rigidity, simultaneously Diophantine approximations Citation: Petković B., Classification of Perturbations of Diophantine $\mathbb{Z}^m$ Actions on Tori of Arbitrary Dimension, Regular and Chaotic Dynamics, 2021, vol. 26, no. 6, pp. 700-716 DOI:10.1134/S1560354721060083
 Alì G.,  Bilotta E.,  Chiaravalloti F.,  Pantano P.,  Pezzi O.,  Scuro C.,  Valentini F. Spatiotemporal Pattern Formation in a Ring of Chua’s Oscillators Abstract In the present work the spontaneous dynamics of a ring of $N$ Chua's oscillators, mutually coupled through a resistor $R_c$ in a nearest-neighbor configuration, is investigated numerically for different strengths of the coupling. A transition from periodic to chaotic global dynamics is observed when the coupling decreases below a critical value and complex patterns in the spatiotemporal dynamics of the ring emerge for a small coupling interval after the transition to chaos. The recovered behavior, as well as the value of the critical threshold, appears to be independent of the size of the ring. We also propose an interpretation of this property, which relates the regular synchronized dynamics of the ring to the dynamics of the isolated oscillator. Finally, for the ring of the coupled oscillator, a theoretical wave dispersion relation is calculated and successfully compared with the results of the numerical simulations, analyzed by classical techniques adopted for turbulent flows. Keywords: coupled nonlinear oscillators, Chua oscillator, Patterns formation, ordinary differential equations, bifurcation Citation: Alì G.,  Bilotta E.,  Chiaravalloti F.,  Pantano P.,  Pezzi O.,  Scuro C.,  Valentini F., Spatiotemporal Pattern Formation in a Ring of Chua’s Oscillators, Regular and Chaotic Dynamics, 2021, vol. 26, no. 6, pp. 717-731 DOI:10.1134/S1560354721060095
 Kudryavtseva E. A.,  Martynchuk N. N. Existence of a Smooth Hamiltonian Circle Action near Parabolic Orbits and Cuspidal Tori Abstract We show that every parabolic orbit of a two-degree-of-freedom integrable system admits a $C^\infty$-smooth Hamiltonian circle action, which is persistent under small integrable $C^\infty$ perturbations. We deduce from this result the structural stability of parabolic orbits and show that they are all smoothly equivalent (in the non-symplectic sense) to a standard model. As a corollary, we obtain similar results for cuspidal tori. Our proof is based on showing that every symplectomorphism of a neighbourhood of a parabolic point preserving the first integrals of motion is a Hamiltonian whose generating function is smooth and constant on the connected components of the common level sets. Keywords: Liouville integrability, parabolic orbit, circle action, structural stability, normal forms Citation: Kudryavtseva E. A.,  Martynchuk N. N., Existence of a Smooth Hamiltonian Circle Action near Parabolic Orbits and Cuspidal Tori, Regular and Chaotic Dynamics, 2021, vol. 26, no. 6, pp. 732-741 DOI:10.1134/S1560354721060101
 Rejeb K. Strongly Reversible Flows on Connected Manifolds Abstract Let $G = \{h_t \ | \ t \in \mathbb R\}$ be a flow of homeomorphisms of a connected $n$-manifold and let $L(G)$ be its limit set. The flow $G$ is said to be strongly reversed by a reflection $R$ if $h_{-t} = R h_t R$ for all $t \in \mathbb R$. In this paper, we study the dynamics of positively equicontinuous strongly reversible flows. If $L(G)$ is nonempty, we discuss the existence of symmetric periodic orbits, and for $n=3$ we prove that such flows must be periodic. If $L(G)$ is empty, we show that $G$ positively equicontinuous implies $G$ strongly reversible and $G$ strongly reversible implies $G$ parallelizable with global section the fixed point set $Fix(R)$. Keywords: strongly reversible, flow of homeomorphisms, positively equicontinuous, periodic orbit, parallelizable, limit set Citation: Rejeb K., Strongly Reversible Flows on Connected Manifolds, Regular and Chaotic Dynamics, 2021, vol. 26, no. 6, pp. 742-755 DOI:10.1134/S1560354721060113
 Bessa M.,  Lopes-Dias J.,  Torres M. Expansiveness and Hyperbolicity in Convex Billiards Abstract We say that a convex planar billiard table $B$ is $C^2$-stably expansive on a fixed open subset $U$ of the phase space if its billiard map $f_B$ is expansive on the maximal invariant set $\Lambda_{B,U}=\bigcap_{n\in\mathbb{Z}}f^n_B(U)$, and this property holds under $C^2$-perturbations of the billiard table. In this note we prove for such billiards that the closure of the set of periodic points of $f_B$ in $\Lambda_{B,U}$ is uniformly hyperbolic. In addition, we show that this property also holds for a generic choice among billiards which are expansive. Keywords: convex planar billiards, hyperbolic sets, expansiveness Citation: Bessa M.,  Lopes-Dias J.,  Torres M., Expansiveness and Hyperbolicity in Convex Billiards, Regular and Chaotic Dynamics, 2021, vol. 26, no. 6, pp. 756-762 DOI:10.1134/S1560354721060125
 Haigh D.,  Katsanikas M.,  Agaoglou M.,  Wiggins S. The Time Evolution of the Trajectories After the Selectivity in a Symmetric Potential Energy Surface with a Post-transition-state Bifurcation Abstract Selectivity is an important phenomenon in chemical reaction dynamics. This can be quantified by the branching ratio of the trajectories that visit one or the other well to the total number of trajectories in a system with a potential with two sequential index-1 saddles and two wells (top well and bottom well). In our case, the relative branching ratio is 1:1 because of the symmetry of our potential energy surface. The mechanisms of transport and the behavior of the trajectories in this kind of systems have been studied recently. In this paper we study the time evolution after the selectivity as energy varies using periodic orbit dividing surfaces. We investigate what happens after the first visit of a trajectory to the region of the top or the bottom well for different values of energy. We answer the natural question: What is the destiny of these trajectories? Keywords: phase space structure, dividing surfaces, chemical physics, periodic orbits, homoclinic and heteroclinic orbits Citation: Haigh D.,  Katsanikas M.,  Agaoglou M.,  Wiggins S., The Time Evolution of the Trajectories After the Selectivity in a Symmetric Potential Energy Surface with a Post-transition-state Bifurcation, Regular and Chaotic Dynamics, 2021, vol. 26, no. 6, pp. 763-774 DOI:10.1134/S1560354721060137
 Kilin A. A.,  Pivovarova E. N. A Particular Integrable Case in the Nonautonomous Problem of a Chaplygin Sphere Rolling on a Vibrating Plane Abstract In this paper we investigate the motion of a Chaplygin sphere rolling without slipping on a plane performing horizontal periodic oscillations. We show that in the system under consideration the projections of the angular momentum onto the axes of the fixed coordinate system remain unchanged. The investigation of the reduced system on a fixed level set of first integrals reduces to analyzing a three-dimensional period advance map on $SO(3)$. The analysis of this map suggests that in the general case the problem considered is nonintegrable. We find partial solutions to the system which are a generalization of permanent rotations and correspond to nonuniform rotations about a body- and space-fixed axis. We also find a particular integrable case which, after time is rescaled, reduces to the classical Chaplygin sphere rolling problem on the zero level set of the area integral. Keywords: Chaplygin sphere, rolling motion, nonholonomic constraint, nonautonomous dynamical system, periodic oscillations, permanent rotations, integrable case, period advance map Citation: Kilin A. A.,  Pivovarova E. N., A Particular Integrable Case in the Nonautonomous Problem of a Chaplygin Sphere Rolling on a Vibrating Plane, Regular and Chaotic Dynamics, 2021, vol. 26, no. 6, pp. 775-786 DOI:10.1134/S1560354721060149

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