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# Volume 26, Number 3, 2021

 Ivanov A. V. On Singularly Perturbed Linear Cocycles over Irrational Rotations Abstract We study a linear cocycle over the irrational rotation $\sigma_{\omega}(x) = x + \omega$ of the circle~$\mathbb{T}^{1}$. It is supposed that the cocycle is generated by a $C^{2}$-map $A_{\varepsilon}: \mathbb{T}^{1} \to SL(2, \mathbb{R})$ which depends on a small parameter $\varepsilon\ll 1$ and has the form of the Poincar\'e map corresponding to a singularly perturbed Hill equation with quasi-periodic potential. Under the assumption that the norm of the matrix $A_{\varepsilon}(x)$ is of order $\exp(\pm \lambda(x)/\varepsilon)$, where $\lambda(x)$ is a positive function, we examine the property of the cocycle to possess an exponential dichotomy (ED) with respect to the parameter~$\varepsilon$. We show that in the limit $\varepsilon\to 0$ the cocycle typically'' exhibits ED only if it is exponentially close to a constant cocycle. Conversely, if the cocycle is not close to a constant one, it does not possess ED, whereas the Lyapunov exponent is typically'' large. Keywords: exponential dichotomy, Lyapunov exponent, reducibility, linear cocycle Citation: Ivanov A. V., On Singularly Perturbed Linear Cocycles over Irrational Rotations, Regular and Chaotic Dynamics, 2021, vol. 26, no. 3, pp. 205-221 DOI:10.1134/S1560354721030011
 Lara R.,  Bengochea A. A Restricted Four-body Problem for the Figure-eight Choreography Abstract In this work we introduce a planar restricted four-body problem where a massless particle moves under the gravitational influence due to three bodies following the figure-eight choreography, and explore some symmetric periodic orbits of this system which turns out to be nonautonomous. We use reversing symmetries to study both theoretically and numerically a certain type of symmetric periodic orbits of this system. The symmetric periodic orbits (initial conditions) were computed by solving some boundary-value problems. Keywords: figure-eight solution, restricted four-body problem, reversing symmetry, periodic orbits, stability Citation: Lara R.,  Bengochea A., A Restricted Four-body Problem for the Figure-eight Choreography, Regular and Chaotic Dynamics, 2021, vol. 26, no. 3, pp. 222-235 DOI:10.1134/S1560354721030023
 Zhu S. Compactness and Index of Ordinary Central Configurations for the Curved $N$-Body Problem Abstract For the curved $n$-body problem, we show that the set of ordinary central configurations is away from singular configurations in $\mathbb{H}^3$ with positive momentum of inertia, and away from a subset of singular configurations in $\mathbb{S}^3$. We also show that each of the $n!/2$ geodesic ordinary central configurations for $n$ masses has Morse index $n-2$. Then we get a direct corollary that there are at least $\frac{(3n-4)(n-1)!}{2}$ ordinary central configurations for given $n$ masses if all ordinary central configurations of these masses are nondegenerate. Keywords: curved $n$-body problem, ordinary central configurations, geodesic configurations, Morse index, compactness, relative equilibrium, hyperbolic relative equilibrium Citation: Zhu S., Compactness and Index of Ordinary Central Configurations for the Curved $N$-Body Problem, Regular and Chaotic Dynamics, 2021, vol. 26, no. 3, pp. 236-257 DOI:10.1134/S1560354721030035
 Kurakin L. G.,  Kurdoglyan A. V. On the Isolation/Nonisolation of a Cosymmetric Equilibrium and Bifurcations in its Neighborhood Abstract A dynamical system with a cosymmetry is considered. V. I. Yudovich showed that a noncosymmetric equilibrium of such a system under the conditions of the general position is a member of a one-parameter family. In this paper, it is assumed that the equilibrium is cosymmetric, and the linearization matrix of the cosymmetry is nondegenerate. It is shown that, in the case of an odd-dimensional dynamical system, the equilibrium is also nonisolated and belongs to a one-parameter family of equilibria. In the even-dimensional case, the cosymmetric equilibrium is, generally speaking, isolated. The Lyapunov – Schmidt method is used to study bifurcations in the neighborhood of the cosymmetric equilibrium when the linearization matrix has a double kernel. The dynamical system and its cosymmetry depend on a real parameter. We describe scenarios of branching for families of noncosymmetric equilibria. Keywords: dynamical system, equilibrium, cosymmetry, bifurcation, Lyapunov – Schmidt method Citation: Kurakin L. G.,  Kurdoglyan A. V., On the Isolation/Nonisolation of a Cosymmetric Equilibrium and Bifurcations in its Neighborhood, Regular and Chaotic Dynamics, 2021, vol. 26, no. 3, pp. 258-270 DOI:10.1134/S1560354721030047
 Kudryashov N. A. Lax Pairs and Rational Solutions of Similarity Reductions for Kupershmidt and Sawada – Kotera Hierarchies Abstract Self-similar reductions for equations of the Kupershmidt and Sawada – Kotera hierarchies are considered. Algorithms for constructing a Lax pair for equations of these hierarchies are presented. Lax pairs for ordinary differential equations of the fifth, seventh and eleventh orders corresponding to the Kupershmidt and the Sawada – Kotera hierarchies are given. The Lax pairs allow us to solve these equations by means of the inverse monodromy transform method. The application of the Painlevé test to the seventh order of the similarity reduction for the Kupershmidt hierarchy is demonstrated. It is shown that special solutions of the similarity reductions for the Kupershnmidt and Sawada – Kotera hierarchies are determined via the transcendents of the $K_1$ and $K_2$ hierarchies. Rational solutions of the similarity reductions of the modified Kupershmidt and Sawada – Kotera hierarchies are given. Special polynomials associated with the self-similar reductions of the Kupershmidt and Sawada – Kotera hierarchies are presented. Rational solutions of some hierarchies are calculated by means of the Miura transformations and taking into account special polynomials. Keywords: higher-order Painlevé equation, Kupershmidt hierarchy, Sawada – Kotera hierarchy, self-similar reduction, special polynomial, special solution Citation: Kudryashov N. A., Lax Pairs and Rational Solutions of Similarity Reductions for Kupershmidt and Sawada – Kotera Hierarchies, Regular and Chaotic Dynamics, 2021, vol. 26, no. 3, pp. 271-292 DOI:10.1134/S1560354721030059
 Rocha A. A. Generic Properties of Mañé's Set of Exact Magnetic Lagrangians Abstract Let $M$ be a closed manifold and $L$ an exact magnetic Lagrangian. In this paper we prove that there exists a residual set $\mathcal{G}$ of $% H^{1}\left( M;\mathbb{R}\right)$ such that the property \begin{equation*} {\widetilde{\mathcal{M}}}\left( c\right) ={\widetilde{\mathcal{A}}}\left( c\right) ={\widetilde{\mathcal{N}}}\left( c\right), \forall c\in \mathcal{G}, \end{equation*} with ${\widetilde{\mathcal{M}}}\left( c\right)$ supporting a uniquely ergodic measure, is generic in the family of exact magnetic Lagrangians. We also prove that, for a fixed cohomology class $c$, there exists a residual set of exact magnetic Lagrangians such that, when this unique measure is supported on a periodic orbit, this orbit is hyperbolic and its stable and unstable manifolds intersect transversally. This result is a version of an analogous theorem, for Tonelli Lagrangians, proven in [6]. Keywords: exact magnetic Lagrangian, Mañé set, genericity Citation: Rocha A. A., Generic Properties of Mañé's Set of Exact Magnetic Lagrangians, Regular and Chaotic Dynamics, 2021, vol. 26, no. 3, pp. 293-304 DOI:10.1134/S1560354721030060
 Nakamura G.,  Grammaticos B.,  Badoual M. Recruitment Effects on the Evolution of Epidemics in a Simple SIR Model Abstract We analyse the patterns of the current epidemic evolution in various countries with the help of a simple SIR model. We consider two main effects: climate induced seasonality and recruitment. The latter is introduced as a way to palliate for the absence of a spatial component in the SIR model. In our approach we mimic the spatial evolution of the epidemic through a gradual introduction of susceptible individuals. We apply our model to the case of France and Australia and explain the appearance of two temporally well-separated epidemic waves. We examine also Brazil and the USA, which present patterns very different from those of the European countries. We show that with our model it is possible to reproduce the observed patterns in these two countries thanks to simple recruitment assumptions. Finally, in order to show the power of the recruitment approach, we simulate the case of the 1918 influenza epidemic reproducing successfully the, by now famous, three epidemic peaks. Keywords: epidemic, modelling, SIR model, seasonality, recruitment Citation: Nakamura G.,  Grammaticos B.,  Badoual M., Recruitment Effects on the Evolution of Epidemics in a Simple SIR Model, Regular and Chaotic Dynamics, 2021, vol. 26, no. 3, pp. 305-319 DOI:10.1134/S1560354721030072

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