Periodic orbits of the restricted three-body problem

We prove, using a variational formulation, the existence of an infinity of periodic solutions of the restricted three-body problem. When the problem has some additional symmetry (in particular, in the autonomous case), we prove the existence of at least two periodic solutions of minimal period , for every . We also study the bifurcation problem in a neighborhood of each closed orbit of the autonomous restricted three-body problem.

     

Tranversality of the invariant manifolds associated to the Lyapunov family of periodic orbits near L2 in the restricted three-body problem

The restricted three-body problem is considered for values of the Jacobi constant C near the value C₂ associated to the Euler critical point L₂. A Lyapunov family of periodic orbits near L₂, the so-called family (c), is born for C = C₂ and exists for values of C less than C₂. These periodic orbits are hyperbolic. The corresponding invariant manifolds meet transversally along homoclinic orbits. In this paper the variation of the transversality is analyzed as a function of the Jacobi constant C and of the mass parameter μ. Asymptotical expressions of the invariant manifolds for C ? C₂ and μ ? 0 are found. Several numerical experiments provide accurate information for the manifolds and a good agreement is found with the asymptotical expressions. Symbolic dynamic techniques are used to show the existence of a large class of motions. In particular the existence of orbits passing in a random way (in a given sense) from the region near one primary to the region near the other is proved.     

On the manifolds of total collapse orbits and of completely parabolic orbits for the n-body problem

In this paper we show for most choices of masses in the coplanar n-body problem and for certain masses in the three-dimensional n-body problem that the set of initial conditions leading to complete collapse forms a smooth submanifold in phase space where the dimension depends upon properties of the limiting configuration. A similar statement holds for completely parabolic motion. By a proper scaling, these two motions become dual to each other in the sense that one forms the unstable set while the other forms the stable set of a particular set of points in the new phase space. Most of the paper is devoted to solving the Painleve-Wintner problem which asserts that these types of orbits cannot enter in an infinite spin.     

Periodic orbits for anisotropic perturbations of the Kepler problem

We study the Kepler problem perturbed by an anisotropic term, that is a potential conformed by a Newtonian term, 1/r$1/r$, plus an anisotropic term, b/(r²[1+?cos²θ])^β/2$b/{\left(r^2[1+?{cos}^2\theta ]\right)}^{\beta /2}$. Because of the anisotropic term, although the system is conservative the angular momentum is not a constant of motion.     

Hyperbolicity of elliptic Lagrangian orbits in the planar three body problem

The linear stability of Lagrangian elliptic equilateral triangle homographic solutions in the classical planar three body problem depends on the mass parameterβ=27(m1m2+m2m3+m3m1)/(m1+m2+m3)2∈[0,9]and the eccentricity e∈[0,1).In this paper we use Maslov-type index to study the stability of these solutions and prove that the elliptic Lagrangian solutions is hyperbolic forβ8 with any eccentricity.     

On a problem on minorants of Lyapunov exponents

We construct examples of families of linear systems whose coefficients continuously depend on the parameter but whose sets of points of lower semicontinuity of the Lyapunov exponents, as well as of their majorants and minorants, treated as functions of the parameter are empty.     

Periodic orbits in the four body problem with large and small masses

This paper studies the dynamics of the four body problem as a limiting system with two of the masses tending to zero. The relative mass and separation of the two small bodies gives three possible limiting problems, the most interesting of which is called the (2+2)-body problem. Similar limits have been considered by previous authors to study the stability of relative equilibrium. In this paper families of periodic orbits are studied, emanating from relative equilibria and from infinity.     

The Tricomi problem on the existence of homoclinic orbits in dissipative systems

The principles of the proof of the existence of homoclinic orbits in dissipative dynamical systems are described. The application of these principles in the case of a Lorenz system enables new criteria for the existence of homoclinic orbits to be formulated.     

Development of the Lyapunov function method in the stability problem for functional-differential equations

In the present paper, we consider the stability problem for delay functional-differential equations with finite delay. We suggest a development of the Lyapunov function method involving the use of scalar comparison equations and limit functions and equations. We prove a localization theorem for the positive limit set of a bounded solution and a theorem on the asymptotic stability of the zero solution. We present examples of sufficient conditions for the asymptotic stability of solutions of systems of the first, second, and arbitrary orders.     

Lyapunov inequalities for the periodic boundary value problem at higher eigenvalues

This paper is devoted to provide some new results on Lyapunov type inequalities for the periodic boundary value problem at higher eigenvalues. Our main result is derived from a detailed analysis on the number and distribution of zeros of nontrivial solutions and their first derivatives, together with the study of some special minimization problems. This allows to obtain the optimal constants. Our applications include the Hill's equation where we give some new conditions on its stability properties and also the study of periodic and nonlinear problems at resonance where we show some new conditions which allow to prove the existence and uniqueness of solutions.