The restricted rhomboidal five-body problem (RRFBP) is a problem in which four positive masses, called the primaries, move two by two in circular motions such that their configuration is always a rhombus, the fifth mass being small and not influencing the motion of the four primaries. In our model, we assume that the fifth mass is in the same plane of the primaries and that the masses of the primaries are m1 = m2 = m and and the radius associated with the circular motion of m1 and m2 is and the one for the masses of m3 and m4 is 1. Similar to the circular restricted three-body problem, we obtain the first integral of motion. The Hamiltonian function which governs the motion of the fifth mass is obtained and has two degrees of freedom depending periodically on time. We use a synodical system of coordinates to eliminate the time dependence. With the help of the Hamiltonian structure, we characterize the regions of possible motion. We show the existence of equilibrium solutions along the coordinate axis as well as off them. We verify that the number of equilibria depends on λ and there can be 11, 13 or 15 equilibrium solutions all unstable. We prove the existence of periodic solutions with short as well as long period. Also we prove the existence of transversal ejection–collision orbits (binary collisions) for certain large values of the Jacobi constant, for an uncountable number of invariant punctured tori in the corresponding energy surface.

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We study the problem in which N bodies, called primaries, of equal masses m are describing circular keplerian solutions in the xy plane and a body µ, of zero mass, moves on a line perpendicular to the plane of motion of the primaries and passing through their center of mass. We show that such a problem is equivalent to the Classical Circular Sitnikov Problem, in which N = 2 and m = 1 2 . We also show that the main parameter in searching for periodic solutions is M = mN , the total mass of all the primaries. We add an analytic study of the period, T (h), as a function of the negative energy h. We generalize some results of [2] and we show the dependence of T (h) on the mass parameter M . Finally, we confirm, the expected result that the case of the Newtonian potential for a homogeneous circular ring of mass M is just the limit case of the problem we have studied, in which we let N go to infinity, while keeping the product mN finite.

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The goal of this work is to analyse the effects of control policies for the coronavirus (COVID-19) epidemic in Brazil. This is done by considering an age-structured SEIR model with a quarantine class and two types of controls. The first one studies the sensitivity with regard to the parameters of the basic reproductive number R 0 which is calculated by a next generation method. The second one evaluates different quarantine strategies by comparing their relative total number of deaths.

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We consider a system of five mass points r1, r2, r3, and r4 with masses m1 = m2 = m and \documentclass[12pt]{minimal}\begin{document}$m_3= m_4= \tilde{m}$\end{document}m3=m4=m̃ moving about a single massive body r0 with mass m0 at its center which is assumed to be the origin of the coordinates system. We assume that the central body r0 makes a generalized force on the four mass points and that such a force is generated by a Manev's type potential, i.e., characterized by a potential of the form \documentclass[12pt]{minimal}\begin{document}$\frac{1}{r}+\frac{\epsilon }{r^2}$\end{document}1r+εr2, on the other hand is assumed that the attraction between the bodies r1, r2, r3, and r4 is of the Newtonian type. This model represents several cases, for instance, when the central body is a spheroid or a radiating source. First, we prove the existence of three different relative rhomboidal solutions, and its (central) configuration is as follows: (1) the rhombus is a square and all primaries have equal masses; (2) the rhombus is not a square but all masses are equal; and (3) the rhombus is not a square and the pairs of primaries have different masses. The first two cases present two parameters: ε, the radiation or the oblateness coefficient and m = μ the common mass of the primaries. In the third case, the only parameter to be considered is ε since it can be shown that in this case both m and \documentclass[12pt]{minimal}\begin{document}$\tilde{m}$\end{document}m̃ depend on ε. Second, we study the stability of these solutions. We determine the values of the parameters (ε, μ) for which the square solution is spectrally stable and we prove that the rhomboidal solutions are unstable in the Lyapunov sense in the other two situations.

Using cutoff functions and periodic extensions, we prove that the Melnikov homoclinic method gives a criterium to show that for a finite time interval [−T,T], with T arbitrarily large, the perturbed system is conjugated to a chaotic one for quite general classes of perturbation functions. The method is applied to specific perturbations of the pendulum and of the Gylden’s problem.