Volume 25, Number 6
Volume 25, Number 6, 2020
Nakamura G., Grammaticos B., Badoual M.
Confinement Strategies in a Simple SIR Model
Abstract
We propose a simple deterministic, differential equationbased, SIR model in order
to investigate the impact of various confinement strategies on a most virulent epidemic. Our
approach is motivated by the current COVID19 pandemic. The main hypothesis is the existence
of two populations of susceptible persons, one which obeys confinement and for which the
infection rate does not exceed 1, and a population which, being non confined for various
imperatives, can be substantially more infective. The model, initially formulated as a differential
system, is discretised following a specific procedure, the discrete system serving as an integrator
for the differential one. Our model is calibrated so as to correspond to what is observed in the
COVID19 epidemic, for the period from February 19 to April 16.
Several conclusions can be reached, despite the very simple structure of our model. First, it is not possible to pinpoint the genesis of the epidemic by just analysing data from when the epidemic is in full swing. It may well turn out that the epidemic has reached a sizeable part of the world months before it became noticeable. Concerning the confinement scenarios, a universal feature of all our simulations is that relaxing the lockdown constraints leads to a rekindling of the epidemic. Thus, we sought the conditions for the second epidemic peak to be lower than the first one. This is possible in all the scenarios considered (abrupt or gradualexit, the latter having linear and stepwise profiles), but typically a gradual exit can start earlier than an abrupt one. However, by the time the gradual exit is complete, the overall confinement times are not too different. From our results, the most promising strategy is that of a stepwise exit. Its implementation could be quite feasible, with the major part of the population (perhaps, minus the fragile groups) exiting simultaneously, but obeying rigorous distancing constraints. 
MazrooeiSebdani R., Hakimi E.
Nondegenerate Hamiltonian Hopf Bifurcations in $\omega : 3 : 6$ Resonance $(\omega = 1$ or $2)$
Abstract
This paper deals with the analysis of Hamiltonian Hopf bifurcations in threedegree
offreedom systems, for which the frequencies of the linearization of the corresponding Hamiltonians are in $\omega : 3 : 6$ resonance $(\omega = 1$ or $2)$. We obtain the truncated secondorder normal form that is not integrable and expressed in terms of the invariants of the reduced phase space. The truncated firstorder normal form gives rise to an integrable system that is analyzed using a reduction to a onedegreeoffreedom system. In this paper, some detuning parameters are considered and nondegenerate Hamiltonian Hopf bifurcations are found. To study Hamiltonian Hopf bifurcations, we transform the reduced Hamiltonian into standard form. 
Kudryashov N. A.
Highly Dispersive Optical Solitons of an Equation with Arbitrary Refractive Index
Abstract
A nonlinear fourthorder differential equation with arbitrary refractive index for
description of the pulse propagation in an optical fiber is considered. The Cauchy problem for
this equation cannot be solved by the inverse scattering transform and we look for solutions
of the equation using the traveling wave reduction. We present a novel method for finding
soliton solutions of nonlinear evolution equations. The essence of this method is based on the
hypothesis about the possible type of an auxiliary equation with an already known solution. This
new auxiliary equation is used as a basic equation to look for soliton solutions of the original
equation. We have found three forms of soliton solutions of the equation at some constraints
on parameters of the equation.

Bogoyavlenskij O. I., Peng Y.
Autotransforms for Some Differential Equations and Their Physical Applications
Abstract
General constructions of differential equations with autotransforms are presented.
Differential operators connecting a linear case of the Grad – Shafranov equation and the axisymmetric Helmholtz equation are found. Infinite families of exact solutions to both equations are derived. 
Sailer S., Eugster S. R., Leine R. I.
The Tippedisk: a Tippetop Without Rotational Symmetry
Abstract
The aim of this paper is to introduce the tippedisk to the theoretical mechanics community as a new mechanicalmathematical archetype for friction induced instability phenomena. We discuss the modeling and simulation of the tippedisk, which is an inhomogeneous disk showing an inversion phenomenon similar but more complicated than the tippetop. In particular, several models with different levels of abstraction, parameterizations and force laws are introduced. Moreover, the numerical simulations are compared qualitatively with recordings from a highspeed camera. Unlike the tippetop, the tippedisk has no rotational symmetry, which greatly complicates the threedimensional nonlinear kinematics. The governing differential equations, which are presented here in full detail, describe all relevant physical effects and serve as a starting point for further research.

Shoptrajanov M., Shekutkovski N.
Shapeinvariant Neighborhoods of Nonsaddle Sets
Abstract
Asymptotically stable attractors are only a particular case of a large family of
invariant compacta whose global topological structure is regular. We devote this paper to
investigating the shape properties of this class of compacta, the nonsaddle sets. Stable attractors
and unstable attractors having only internal explosions are examples of nonsaddle sets. The
main aim of this paper is to generalize the wellknown theorem for the shape of attractors to
nonsaddle sets using the intrinsic approach to shape which combines continuity up to a covering
and the corresponding homotopies of first order.

Shepelev I. A., Bukh A. V., Muni S. S., Anishchenko V. S.
Quantifying the Transition from Spiral Waves to Spiral Wave Chimeras in a Lattice of Selfsustained Oscillators
Abstract
The present work is devoted to the detailed quantification of the transition from spiral waves to spiral wave chimeras in a network of selfsustained oscillators with twodimensional geometry. The basic elements of the network under consideration are the van der Pol oscillator or the FitzHugh – Nagumo neuron. Both of the models are in the regime of relaxation oscillations. We analyze the regime by using the indices of local sensitivity, which enables us to evaluate the sensitivity of each oscillator at a finite time. Spiral waves are observed in both lattices when the interaction between elements has a local character. The dynamics of all the elements is regular. There are no pronounced highsensitive regions. We have discovered that, when the coupling becomes nonlocal, the features of the system change significantly. The oscillation regime of the spiral wave center element switches to a chaotic one. Besides, a region with high sensitivity occurs around the wave center oscillator. Moreover, we show that the latter expands in space with elongation of the coupling range. As a result, an incoherence cluster of the spiral wave chimera is formed exactly within this highsensitive area. A sharp increase in the values of the maximal Lyapunov exponent in the positive region leads to the formation of the incoherence cluster. Furthermore, we find that the system can even switch to a hyperchaotic regime when several Lyapunov exponents become positive.

Xu J., You J.
Persistence of Hyperbolictype Degenerate Lowerdimensional Invariant Tori with Prescribed Frequencies in Hamiltonian Systems
Abstract
It is known that under Kolmogorov’s nondegeneracy condition, the nondegenerate
hyperbolic invariant torus with Diophantine frequencies will persist under small perturbations,
meaning that the perturbed system still has an invariant torus with prescribed frequencies.
However, the degenerate torus is sensitive to perturbations. In this paper, we prove the
persistence of two classes of hyperbolictype degenerate lowerdimensional invariant tori, one
of them corrects an earlier work [34] by the second author. The proof is based on a modified
KAM iteration and analysis of stability of degenerate critical points of analytic functions.

Yuan P., Llibre J.
Tangential Trapezoid Central Configurations
Abstract
A tangential trapezoid, also called a circumscribed trapezoid, is a trapezoid whose
four sides are all tangent to a circle within the trapezoid: the incircle or inscribed circle. In
this paper we classify all planar fourbody central configurations, where the four bodies are at
the vertices of a tangential trapezoid.

Talamucci F.
Rheonomic Systems with Nonlinear Nonholonomic Constraints: The Voronec Equations
Abstract
One of the earliest formulations of dynamics of nonholonomic systems traces back
to 1895 and is due to Chaplygin, who developed his analysis under the assumption that a certain
number of the generalized coordinates do not occur either in the kinematic constraints or in
the Lagrange function. A few years later Voronec derived equations of motion for nonholonomic
systems removing the restrictions demanded by the Chaplygin systems. Although the methods
encountered in the following years favor the use of the quasicoordinates, we will pursue the
Voronec method, which deals with the generalized coordinates directly. The aim is to establish
a procedure for extending the equations of motion to nonlinear nonholonomic systems, even in
the rheonomic case.

Kozlov V. V.
Nonequilibrium Statistical Mechanics of Weakly Ergodic Systems
Abstract
The properties of the Gibbs ensembles of Hamiltonian systems describing the
motion along geodesics on a compact configuration manifold are discussed.We introduce weakly
ergodic systems for which the time average of functions on the configuration space is constant
almost everywhere. Usual ergodic systems are, of course, weakly ergodic, but the converse is not
true. A range of questions concerning the equalization of the density and the temperature of a
Gibbs ensemble as time increases indefinitely are considered. In addition, the weak ergodicity
of a billiard in a rectangular parallelepiped with a partition wall is established.

Artemova E. M., Karavaev Y. L., Mamaev I. S., Vetchanin E. V.
Dynamics of a Spherical Robot with Variable Moments of Inertia and a Displaced Center of Mass
Abstract
The motion of a spherical robot with periodically changing moments of inertia,
internal rotors and a displaced center of mass is considered. It is shown that, under some
restrictions on the displacement of the center of mass, the system of interest features chaotic
dynamics due to separatrix splitting. A stability analysis is made of the upper equilibrium
point of the ball and of the periodic solution arising in its neighborhood, in the case of periodic
rotation of the rotors. It is shown that the lower equilibrium point can become unstable in the
case of fixed rotors and periodically changing moments of inertia.

Borisov A. V., Ivanov A. P.
Dynamics of the Tippe Top on a Vibrating Base
Abstract
This paper studies the conditions under which the tippe top inverts in the presence
of vibrations of the base along the vertical. A vibrational potential is constructed by averaging
and it is shown that, when this potential is added to the system, the Jellett integral is preserved.
This makes it possible to apply the modified Routh method and to find the effective potential
to whose critical points permanent rotations or regular precessions of the tippe top correspond.
Tippe top inversion is possible for a sufficiently large initial angular velocity under the condition
that spinning with the lowest position of the center of gravity is unstable, spinning with the
highest position of the center of gravity is stable, and that there are no precessions. Cases are
found in which there is no inversion in the absence of vibrations, but it can be brought about
by a suitable choice of the mean value of the squared velocity of the base. In particular, this
type includes a ball with a spherical cavity filled with a denser substance.

Kruglov V., Malyshev D. S., Pochinka O. V., Shubin D. D.
On Topological Classification of Gradientlike Flows on an $n$sphere in the Sense of Topological Conjugacy
Abstract
In this paper, we study gradientlike flows without heteroclinic intersections on an
$n$sphere up to topological conjugacy. We prove that such a flow is completely defined by a
bicolor tree corresponding to a skeleton formed by codimension one separatrices. Moreover, we
show that such a tree is a complete invariant for these flows with respect to the topological
equivalence also. This result implies that for these flows with the same (up to a change
of coordinates) partitions into trajectories, the partitions for elements, composing isotopies
connecting timeone shifts of these flows with the identity map, also coincide. This phenomenon
strongly contrasts with the situation for flows with periodic orbits and connections, where
one class of equivalence contains continuum classes of conjugacy. In addition, we realize every
connected bicolor tree by a gradientlike flow without heteroclinic intersections on the $n$sphere.
In addition, we present a lineartime algorithm on the number of vertices for distinguishing these
trees.

Kilin A. A., Pivovarova E. N.
Stability and Stabilization of Steady Rotations of a Spherical Robot on a Vibrating Base
Abstract
This paper addresses the problem of a spherical robot having an axisymmetric pendulum drive and rolling without slipping on a vibrating plane. It is shown that this system admits partial solutions (steady rotations) for which the pendulum rotates about its vertical symmetry axis. Special attention is given to problems of stability and stabilization of these solutions. An analysis of the constraint reaction is performed, and parameter regions are identified in which a stabilization of the spherical robot is possible without it losing contact with the plane. It is shown that the partial solutions can be stabilized by varying the angular velocity of rotation of the pendulum about its symmetry axis, and that the rotation of the pendulum is a necessary condition for stabilization without the robot losing contact with the plane.
