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# 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 equation-based, SIR model in order to investigate the impact of various confinement strategies on a most virulent epidemic. Our approach is motivated by the current COVID-19 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 COVID-19 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. Keywords: epidemics, modelling, SIR model, lockdown Citation: Nakamura G.,  Grammaticos B.,  Badoual M., Confinement Strategies in a Simple SIR Model, Regular and Chaotic Dynamics, 2020, vol. 25, no. 6, pp. 509-521 DOI:10.1134/S1560354720060015
 Mazrooei-Sebdani 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- of-freedom 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 second-order normal form that is not integrable and expressed in terms of the invariants of the reduced phase space. The truncated first-order normal form gives rise to an integrable system that is analyzed using a reduction to a one-degree-of-freedom 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. Keywords: Hamiltonian $\omega : 3 : 6$ resonance $(\omega = 1$ or $2)$, integrability, reduction, normal forms, Hamiltonian Hopf bifurcation Citation: Mazrooei-Sebdani R.,  Hakimi E., Nondegenerate Hamiltonian Hopf Bifurcations in $\omega : 3 : 6$ Resonance $(\omega = 1$ or $2)$, Regular and Chaotic Dynamics, 2020, vol. 25, no. 6, pp. 522-536 DOI:10.1134/S1560354720060027
 Kudryashov N. A. Highly Dispersive Optical Solitons of an Equation with Arbitrary Refractive Index Abstract A nonlinear fourth-order 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. Keywords: nonlinear mathematical model, traveling wave, solitary wave, pulse propagation, optical fiber Citation: Kudryashov N. A., Highly Dispersive Optical Solitons of an Equation with Arbitrary Refractive Index, Regular and Chaotic Dynamics, 2020, vol. 25, no. 6, pp. 537-543 DOI:10.1134/S1560354720060039
 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. Keywords: differential operators, Grad – Shafranov equation, force-free plasma equilibria, exact solutions, Helmholtz equation Citation: Bogoyavlenskij O. I.,  Peng Y., Autotransforms for Some Differential Equations and Their Physical Applications, Regular and Chaotic Dynamics, 2020, vol. 25, no. 6, pp. 544-552 DOI:10.1134/S1560354720060040
 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 mechanical-mathematical 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 high-speed camera. Unlike the tippetop, the tippedisk has no rotational symmetry, which greatly complicates the three-dimensional 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. Keywords: gyroscopic system, friction, contact modelling, dynamics, inversion phenomenon Citation: Sailer S.,  Eugster S. R.,  Leine R. I., The Tippedisk: a Tippetop Without Rotational Symmetry, Regular and Chaotic Dynamics, 2020, vol. 25, no. 6, pp. 553-580 DOI:10.1134/S1560354720060052
 Shoptrajanov M.,  Shekutkovski N. Shape-invariant 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 well-known 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. Keywords: shape, intrinsic shape, attractor, nonsaddle set, regular covering, proximate sequence, Lyapunov function Citation: Shoptrajanov M.,  Shekutkovski N., Shape-invariant Neighborhoods of Nonsaddle Sets, Regular and Chaotic Dynamics, 2020, vol. 25, no. 6, pp. 581-596 DOI:10.1134/S1560354720060064
 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 Self-sustained 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 self-sustained 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 high-sensitive 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 high-sensitive 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. Keywords: spatiotemporal pattern, chimera state, van der Pol oscillator, FitzHugh – Nagumo neuron, spiral wave, spiral wave chimera, nonlocal interaction, Lyapunov exponent Citation: 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 Self-sustained Oscillators, Regular and Chaotic Dynamics, 2020, vol. 25, no. 6, pp. 597-615 DOI:10.1134/S1560354720060076
 Xu J.,  You J. Persistence of Hyperbolic-type Degenerate Lower-dimensional 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 hyperbolic-type degenerate lower-dimensional 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. Keywords: Hamiltonian system, KAM iteration, degenerate equilibrium, invariant tori Citation: Xu J.,  You J., Persistence of Hyperbolic-type Degenerate Lower-dimensional Invariant Tori with Prescribed Frequencies in Hamiltonian Systems, Regular and Chaotic Dynamics, 2020, vol. 25, no. 6, pp. 616-650 DOI:10.1134/S1560354720060088
 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 in-circle or inscribed circle. In this paper we classify all planar four-body central configurations, where the four bodies are at the vertices of a tangential trapezoid. Keywords: convex central configuration, four-body problem, tangential trapezoid Citation: Yuan P.,  Llibre J., Tangential Trapezoid Central Configurations, Regular and Chaotic Dynamics, 2020, vol. 25, no. 6, pp. 651-661 DOI:10.1134/S156035472006009X
 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 quasi-coordinates, 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. Keywords: nonholonomic systems, nonlinear constraints, Lagrangian equations of motion Citation: Talamucci F., Rheonomic Systems with Nonlinear Nonholonomic Constraints: The Voronec Equations, Regular and Chaotic Dynamics, 2020, vol. 25, no. 6, pp. 662-673 DOI:10.1134/S1560354720060106
 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. Keywords: Hamiltonian system, Liouville and Gibbs measures, Gibbs ensemble, weak ergodicity, mixing, billiard in a polytope Citation: Kozlov V. V., Nonequilibrium Statistical Mechanics of Weakly Ergodic Systems, Regular and Chaotic Dynamics, 2020, vol. 25, no. 6, pp. 674-688 DOI:10.1134/S1560354720060118
 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. Keywords: nonholonomic constraint, rubber rolling, unbalanced ball, rolling on a plane Citation: 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, Regular and Chaotic Dynamics, 2020, vol. 25, no. 6, pp. 689-706 DOI:10.1134/S156035472006012X
 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. Keywords: tippe top, dry friction, Jellett integral Citation: Borisov A. V.,  Ivanov A. P., Dynamics of the Tippe Top on a Vibrating Base, Regular and Chaotic Dynamics, 2020, vol. 25, no. 6, pp. 707-715 DOI:10.1134/S1560354720060131
 Kruglov V.,  Malyshev D. S.,  Pochinka O. V.,  Shubin D. On Topological Classification of Gradient-like Flows on an $n$-sphere in the Sense of Topological Conjugacy Abstract In this paper, we study gradient-like 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 time-one 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 gradient-like flow without heteroclinic intersections on the $n$-sphere. In addition, we present a linear-time algorithm on the number of vertices for distinguishing these trees. Keywords: gradient-like flow, topological classification, topological conjugacy, $n$-sphere, lineartime algorithm Citation: Kruglov V.,  Malyshev D. S.,  Pochinka O. V.,  Shubin D., On Topological Classification of Gradient-like Flows on an $n$-sphere in the Sense of Topological Conjugacy, Regular and Chaotic Dynamics, 2020, vol. 25, no. 6, pp. 716-728 DOI:10.1134/S1560354720060143
 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. Keywords: spherical robot, vibrations, stability, stabilization, partial solutions, constraint reaction, Lagrange top, Kapitsa pendulum Citation: Kilin A. A.,  Pivovarova E. N., Stability and Stabilization of Steady Rotations of a Spherical Robot on a Vibrating Base, Regular and Chaotic Dynamics, 2020, vol. 25, no. 6, pp. 729-752 DOI:10.1134/S1560354720060155

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