Claudio Vidal

Restricted rhomboidal five-body problem

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 ? 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.

Stability and global dynamic of a stage-structured predator-prey model with group defense mechanism of the prey

Dynamics of predator-prey model where the predator feeds on two age classes of prey.Existence of defense mechanism and the interference of the prey and the age.The coexistence of the predator and the prey populations is possible.There exists a bounded range of the carrying capacity for co-existence. In this paper we describe the flow of a predator-prey model where the predator feeds on two age classes of prey. Some basic features of the flow are proved under very mild hypothesis on the functional responses. To study the relationship between the ecological parameters and the behavioral ones, we also analyze a special case of the model where modified versions of the Holling type II functional responses are considered to take into account both a certain interference of the age classes of the prey on the predator activity and a defense mechanism of the juvenile class of the prey. Indeed, we find that independently of the profit that each one of the prey age class provides to the predator, the coexistence of the predator and the prey populations is only possible for a bounded range of the carrying capacity due to the joint action of the defense mechanism and the interference of the prey and the age.

Stability of the planar equilibrium solutions of a restricted 1 + N body problem

We started our studies with a planar Eulerian restricted four-body problem (ERFBP) where three masses move in circular orbits such that their configuration is always collinear. The fourth mass is small and does not influence the motion of the three primaries. In our model we assume that one of the primaries has mass 1 and is located at the origin and two masses of size µ rotate around it uniformly. The problem was studied in [3], where it was shown that there exist noncollinear equilibria, which are Lyapunov stable for small values of µ. KAM theory is used to establish the stability of the equilibria. Our computations do not agree with those given in [3], although our conclusions are similar. The ERFBP is a special case of the 1 + N restricted body problem with N = 2. We are able to do the computations for any N and find that the stability results are very similar to those for N = 2. Since the 1 + N body configuration can be stable when N > 6, these results could be of more significance than for the case N = 2.

Stability in a rhomboidal 5-body problem with generalized central forces

We consider a system of five mass points r1, r2, r3, and r4 with masses m1 = m2 = m and m3=m4=m 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 Manevs type potential, i.e., characterized by a potential of the form 1r+er2, 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 pa...

Stability of the Polar Equilibria in a Restricted Three-Body Problem on the Sphere

In this paper we consider a symmetric restricted circular three-body problem on the surface S2 of constant Gaussian curvature κ = 1. This problem consists in the description of the dynamics of an infinitesimal mass particle attracted by two primaries with identical masses, rotating with constant angular velocity in a fixed parallel of radius a ∈ (0, 1). It is verified that both poles of S2 are equilibrium points for any value of the parameter a. This problem is modeled through a Hamiltonian system of two degrees of freedom depending on the parameter a. Using results concerning nonlinear stability, the type of Lyapunov stability (nonlinear) is provided for the polar equilibria, according to the resonances. It is verified that for the north pole there are two values of bifurcation (on the stability)

THE FLOW OF CLASSICAL MECHANICAL CUBIC POTENTIAL SYSTEMS

a = \frac{{\sqrt {4 - \sqrt 2 } }}{2}

Stability of equilibrium solutions in the critical case of even-order resonance in periodic Hamiltonian systems with one degree of freedom

a=4−22 and