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# Volume 25, Number 2, 2020

 Cárcamo-Díaz D.,  Palacián J. F.,  Vidal C.,  Yanguas P. On the Nonlinear Stability of the Triangular Points in the Circular Spatial Restricted Three-body Problem Abstract The well-known problem of the nonlinear stability of $L_4$ and $L_5$ in the circular spatial restricted three-body problem is revisited. Some new results in the light of the concept of Lie (formal) stability are presented. In particular, we provide stability and asymptotic estimates for three specific values of the mass ratio that remained uncovered. Moreover, in many cases we improve the estimates found in the literature. Keywords: restricted three-body problem, $L_4$ and $L_5$, elliptic equilibria, resonances, formal and Lie stability, exponential estimates Citation: Cárcamo-Díaz D.,  Palacián J. F.,  Vidal C.,  Yanguas P., On the Nonlinear Stability of the Triangular Points in the Circular Spatial Restricted Three-body Problem, Regular and Chaotic Dynamics, 2020, vol. 25, no. 2, pp. 131-148 DOI:10.1134/S156035472002001X
 Lim C. C. General Jacobi Coordinates and Herman Resonance for Some Nonheliocentric Celestial $N$-body Problems Abstract The general Jacobi symplectic variables generated by a combinatorial algorithm from the full binary tree $T(N)$ are used to formulate some nonheliocentric gravitational $N$-body problems in perturbation form. The resulting uncoupled term $H_U$ for $(N-1)$ independent Keplerian motions and the perturbation term $H_P$ are both explicitly dependent on the partial ordering induced by the tree $T(N)$. This leads to suitable conditions on separations of the $N$ bodies for the perturbation to be small. We prove the Herman resonance for a new approximation of the 5-body problem. Full details of the derivations of the perturbation form and Herman resonance are given only in the case of five bodies using the caterpillar binary tree $T_c(5)$. Keywords: general Jacobi coordinates, perturbation theory, celestial $N$-body problems, Herman resonances Citation: Lim C. C., General Jacobi Coordinates and Herman Resonance for Some Nonheliocentric Celestial $N$-body Problems, Regular and Chaotic Dynamics, 2020, vol. 25, no. 2, pp. 149-165 DOI:10.1134/S1560354720020021
 Biswas A.,  Kara A. H.,  Zhou  Q.,  Alzahrani A. K.,  Belic M. R. Conservation Laws for Highly Dispersive Optical Solitons in Birefringent Fibers Abstract This paper reports conservation laws for highly dispersive optical solitons in birefringent fibers. Three forms of nonlinearities are studied which are Kerr, polynomial and nonlocal laws. Power, linear momentum and Hamiltonian are conserved for these types of nonlinear refractive index. Keywords: conservation laws, highly dispersive solitons, birefringent fibers Citation: Biswas A.,  Kara A. H.,  Zhou  Q.,  Alzahrani A. K.,  Belic M. R., Conservation Laws for Highly Dispersive Optical Solitons in Birefringent Fibers, Regular and Chaotic Dynamics, 2020, vol. 25, no. 2, pp. 166-177 DOI:10.1134/S1560354720020033
 Grammaticos B.,  Willox R.,  Satsuma J. Revisiting the Human and Nature Dynamics Model Abstract We present a simple model for describing the dynamics of the interaction between a homogeneous population or society, and the natural resources and reserves that the society needs for its survival. The model is formulated in terms of ordinary differential equations, which are subsequently discretised, the discrete system providing a natural integrator for the continuous one. An ultradiscrete, generalised cellular automaton-like, model is also derived. The dynamics of our simple, three-component, model are particularly rich exhibiting either a route to a steady state or an oscillating, limit cycle-type regime or to a collapse. While these dynamical behaviours depend strongly on the choice of the details of the model, the important conclusion is that a collapse or near collapse, leading to the disappearance of the population or to a complete transfiguration of its societal model, is indeed possible. Keywords: population dynamics, dynamical systems, collapse, resources and reserves, discretisation, generalised cellular automaton Citation: Grammaticos B.,  Willox R.,  Satsuma J., Revisiting the Human and Nature Dynamics Model, Regular and Chaotic Dynamics, 2020, vol. 25, no. 2, pp. 178-198 DOI:10.1134/S1560354720020045
 Simó C. Simple Flows on Tori with Uncommon Chaos Abstract We consider a family of simple flows in tori that display chaotic behavior in a wide sense. But these flows do not have homoclinic nor heteroclinic orbits. They have only a fixed point which is of parabolic type. However, the dynamics returns infinitely many times near the fixed point due to quasi-periodicity. A preliminary example is given for maps introduced in a paper containing many examples of strange attractors in [6]. Recently, a family of maps similar to the flows considered here was studied in [9]. In the present paper we consider the case of 2D tori and the extension to tori of arbitrary finite dimension. Some other facts about exceptional frequencies and behavior around parabolic fixed points are also included. Keywords: chaos without homoclinic/heteroclinic points, chaotic flows on tori, the returning role of quasi-periodicity, zero maximal Lyapunov exponents, the role of parabolic points, exceptional frequencies Citation: Simó C., Simple Flows on Tori with Uncommon Chaos, Regular and Chaotic Dynamics, 2020, vol. 25, no. 2, pp. 199-214 DOI:10.1134/S1560354720020057
 Mamaev I. S.,  Vetchanin  E. V. Dynamics of Rubber Chaplygin Sphere under Periodic Control Abstract This paper examines the motion of a balanced spherical robot under the action of periodically changing moments of inertia and gyrostatic momentum. The system of equations of motion is constructed using the model of the rolling of a rubber body (without slipping and twisting) and is nonconservative. It is shown that in the absence of gyrostatic momentum the equations of motion admit three invariant submanifolds corresponding to plane-parallel motion of the sphere with rotation about the minor, middle and major axes of inertia. The abovementioned motions are quasi-periodic, and for the numerical estimate of their stability charts of the largest Lyapunov exponent and charts of stability are plotted versus the frequency and amplitude of the moments of inertia. It is shown that rotations about the minor and major axes of inertia can become unstable at sufficiently small amplitudes of the moments of inertia. In this case, the so-called “Arnol’d tongues” arise in the stability chart. Stabilization of the middle unstable axis of inertia turns out to be possible at sufficiently large amplitudes of the moments of inertia, when the middle axis of inertia becomes the minor axis for a part of a period. It is shown that the nonconservativeness of the system manifests itself in the occurrence of limit cycles, attracting tori and strange attractors in phase space. Numerical calculations show that strange attractors may arise through a cascade of period-doubling bifurcations or after a finite number of torus-doubling bifurcations. Keywords: nonholonomic constraints, rubber rolling, periodic control, stability analysis, perioddoubling bifurcation, torus-doubling bifurcation Citation: Mamaev I. S.,  Vetchanin  E. V., Dynamics of Rubber Chaplygin Sphere under Periodic Control, Regular and Chaotic Dynamics, 2020, vol. 25, no. 2, pp. 215-236 DOI:10.1134/S1560354720020069

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