Volume 27, Number 2
Volume 27, Number 2, 2022
Alexey Borisov Memorial Volume
Treschev D. V.
Isochronicity in 1 DOF
Abstract
Our main result is the complete set of explicit conditions necessary and sufficient for
isochronicity of a Hamiltonian system with one degree of freedom. The conditions are presented
in terms of Taylor coefficients of the Hamiltonian function.

Dragović V., Gasiorek S., Radnović M.
Billiard Ordered Games and Books
Abstract
The aim of this work is to put together two novel concepts from the theory of integrable billiards: billiard ordered games and confocal billiard books. Billiard books appeared recently in the work of Fomenko’s school, in particular, of V.Vedyushkina. These more complex billiard domains are obtained by gluing planar sets bounded by arcs of confocal conics along common edges. Such domains are used in this paper to construct the configuration space for billiard ordered games.We analyse dynamical and topological properties of the systems obtained in that way.

BravoDoddoli A., Montgomery R.
Geodesics in Jet Space
Abstract
The space $J^k$ of $k$jets of a real function of one real variable $x$ admits the structure of Carnot group type. As such, $J^k$ admits
a submetry (subRiemannian submersion) onto the Euclidean plane. Horizontal lifts of Euclidean lines (which
are the left translates of horizontal oneparameter subgroups) are thus globally minimizing geodesics on $J^k$.
All $J^k$geodesics, minimizing or not, are constructed from degree $k$ polynomials in $x$ according to [7–9],
reviewed here.
The constant polynomials correspond to the horizontal lifts of lines. Which other polynomials yield globally minimizers and what
do these minimizers look like? We give a partial answer. Our methods include constructing
an intermediate threedimensional ``magnetic'' subRiemannian space lying between the jet space and the plane, solving a Hamilton – Jacobi (eikonal) equations on this space, and analyzing period asymptotics associated to period degenerations arising from twoparameter families of these polynomials. Along the way, we conjecture the independence of the cut time of any geodesic on jet space from the starting location on that geodesic.

Masroor E., Stremler M. A.
On the Topology of the Atmosphere Advected by a Periodic Array of Axisymmetric Thincored Vortex Rings
Abstract
The fluid motion produced by a spatially periodic array of identical, axisymmetric, thincored vortex rings is investigated.
It is well known that such an array moves uniformly without change of shape or form in the direction of the central axis of symmetry, and is therefore an equilibrium solution of Euler's equations. In a frame of reference moving with the system of vortex rings, the motion of passive fluid particles is investigated as a function of the two nondimensional parameters that define this system: $\varepsilon = a/R$, the ratio of minor radius to major radius of the torusshaped vortex rings, and $\lambda=L/R$, the separation of the vortex rings normalized by their radii. Two bifurcations in the streamline topology are found that depend significantly on $\varepsilon$ and $\lambda$; these bifurcations delineate three distinct shapes of the ``atmosphere'' of fluid particles that move together with the vortex ring for all time. Analogous to the case of an isolated vortex ring, the atmospheres can be ``thinbodied'' or ``thickbodied''. Additionally, we find the occurrence of a ``connected'' system, in which the atmospheres of neighboring rings touch at an invariant circle of fluid particles that is stationary in a frame of reference moving with the vortex rings.

Gonchenko M. S., Kazakov A. O., Samylina E. A., Shykhmamedov A.
On 1:3 Resonance Under Reversible Perturbations of Conservative Cubic Hénon Maps
Abstract
We consider reversible nonconservative perturbations of the conservative cubic Hénon maps $H_3^{\pm}: \bar x = y, \bar y = x + M_1 + M_2 y \pm y^3$ and study their influence on the 1:3 resonance, i.e., bifurcations of fixed points with eigenvalues $e^{\pm i 2\pi/3}$. It
follows from [1] that this resonance
is degenerate for $M_1=0$, $M_2=1$ when the corresponding
fixed point is elliptic. We show that bifurcations of this
point
under reversible perturbations give rise to four 3periodic orbits, two of them are symmetric
and conservative (saddles in the case of map $H_3^+$ and elliptic orbits in the case of map $H_3^$),
the other two orbits are nonsymmetric and they compose symmetric couples of dissipative orbits
(attracting and repelling orbits in the case of map $H_3^+$ and saddles with the Jacobians less
than 1 and greater than 1 in the case of map $H_3^$). We show that these local symmetrybreaking
bifurcations can lead to mixed dynamics due to accompanying global reversible bifurcations of
symmetric nontransversal homo and heteroclinic cycles. We also generalize the results
of [1] to the case of the $p:q$ resonances with odd $q$ and show that
all of them are also degenerate for the
maps $H_3^{\pm}$ with $M_1=0$.

Ovsyannikov I. I.
On the Birth of Discrete Lorenz Attractors Under Bifurcations of 3D Maps with Nontransversal Heteroclinic Cycles
Abstract
Lorenz attractors are important objects in the modern theory of chaos. The reason,
on the one hand, is that they are encountered in various natural applications (fluid dynamics,
mechanics, laser dynamics, etc.). On the other hand, Lorenz attractors are robust in the sense
that they are generally not destroyed by small perturbations (autonomous, nonautonomous,
stochastic). This allows us to be sure that the object observed in the experiment is exactly a
chaotic attractor rather than a longtime periodic orbit.
Discretetime analogs of the Lorenz attractor possess even more complicated structure — they allow homoclinic tangencies of invariant manifolds within the attractor. Thus, discrete Lorenz attractors belong to the class of wild chaotic attractors. These attractors can be born in codimensionthree local and certain global (homoclinic and heteroclinic) bifurcations. While various homoclinic bifurcations leading to such attractors have been studied, for heteroclinic cycles only cases where at least one of the fixed points is a saddlefocus have been considered to date. In the present paper the case of a heteroclinic cycle consisting of saddle fixed points with a quadratic tangency of invariant manifolds is considered. It is shown that, in order to have threedimensional chaos such as the discrete Lorenz attractors, one needs to avoid the existence of lowerdimensional global invariant manifolds. Thus, it is assumed that either the quadratic tangency or the transversal heteroclinic orbit is nonsimple. The main result of the paper is the proof that the original system is the limiting point in the space of dynamical systems of a sequence of domains in which the diffeomorphism possesses discrete Lorenz attractors. 
Agaoglou M., Katsanikas M., Wiggins S.
The Influence of a Parameter that Controls the Asymmetry of a Potential Energy Surface with an Entrance Channel and Two Potential Wells
Abstract
In this paper we study an asymmetric valleyridge inflection point (VRI) potential,
whose energy surface (PES) features two sequential index1 saddles (the upper and the lower),
with one saddle having higher energy than the other, and two potential wells separated by the
lower index1 saddle. We show how the depth and the flatness of our potential changes as we
modify the parameter that controls the asymmetry as well as how the branching ratio (ratio
of the trajectories that enter each well) is changing as we modify the same parameter and its
correlation with the area of the lobes as they have been formed by the stable and unstable
manifolds that have been extracted from the gradient of the LD scalar fields.

Nikishina N. N., Rybalova E. V., Strelkova G. I., Vadivasova T. E.
Destruction of Cluster Structures in an Ensemble of Chaotic Maps with Noisemodulated Nonlocal Coupling
Abstract
We study numerically the spatiotemporal dynamics of a ring network of nonlocally
coupled logistic maps when the coupling strength is modulated by colored Gaussian noise. Two
cases of noise modulation are considered: 1) when the coupling coefficients characterizing the
influence of neighbors on different elements are subjected to independent noise sources, and 2)
when the coupling coefficients for all the network elements are modulated by the same stochastic
signal. Without noise, the ring of chaotic maps exhibits a chimera state. The impact of noisemodulated
coupling between the ring elements is explored when the parameter, which controls
the correlation time and the spectral width of colored noise, and the noise intensity are varied.
We investigate how the spatiotemporal structures observed in the ring evolve as the noise
parameters change. The numerical results obtained are used to construct regime diagrams for
the two cases of noise modulation. Our findings show the possibility of controlling the spatial
structures in the ring in the presence of noise. Depending on the type of noise modulation, the
spectral properties and intensity of colored noise, one can suppress the incoherent clusters of
chimera states, and induce the regime of solitary states or synchronize chaotic oscillations of
all the ring elements.
