Volume 26, Number 6
Volume 26, Number 6, 2021
Special Issue: 200th birthday of Hermann von Helmholtz/Regular Papers
Shashikanth B. N., Kidambi R.
The CentroidDeformation Decomposition for Buoyant Vortex Patch Motion
Abstract
The motion of a twodimensional 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 centroidfixed 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.

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 alonghelix velocity. A metric term related to the pitch of
the helix enters these equations, which reduce to twodimensional and axisymmetric dynamics
in appropriate limits. We take the vorticity and alonghelix velocity component to be piecewise
constant. In addition to this vortex patch, a vortex sheet develops when the alonghelix 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 alonghelix 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.

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 vortexsource interaction,
now coupled with an unsteady wind, leading to a timevarying circulation and source strength.
Firstly, we recover the various fixed points of the two vortexsource 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.

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.

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.

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
vortexfree, 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.

Bogoyavlenskij O. I., Peng Y.
Exact Solutions to the Beltrami Equation with a Nonconstant $\alpha({\mathbf x})$
Abstract
Infinite families of new exact solutions to the Beltrami equation with a nonconstant
$\alpha({\mathbf x})$ are derived. Differential operators connecting the steady axisymmetric Klein – Gordon
equation and a special case of the Grad – Shafranov equation are constructed. A Lie semigroup
of nonlinear transformations of the Grad – Shafranov equation is found.

Petković B.
Classification of Perturbations of Diophantine $\mathbb{Z}^m$ Actions on Tori of Arbitrary Dimension
Abstract
We generalize results of Moser [17] 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 [17] 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.

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
nearestneighbor 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.

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 twodegreeoffreedom 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 nonsymplectic 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.

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 $Fix(R)$.

Bessa M., LopesDias 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.

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 Posttransitionstate 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 index1 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?

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 threedimensional 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 spacefixed 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.
