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We consider the Manev potential in an anisotropic space, i.e., such that the force acts differently in each direction. Using a generalization of the Poincaré continuation method we study the existence of periodic solutions for weak anisotropy. In particular we find that the symmetric periodic orbits of the Manev system are perturbed to periodic orbits in… (More)

Using a completely analytic procedure – based on a suitable extension of a classical method – we discuss an approach to the PoincaréMel’nikov theory, which can be conveniently applied also to the case of non-hyperbolic critical points, and even if the critical point is located at the infinity. In this paper, we concentrate our attention on the latter case,… (More)

Homoclinic chaos is usually examined with the hypothesis of hyperbolicity of the critical point. We consider here, following a (suitably adjusted) classical analytic method, the case of non-hyperbolic points and show that, under a Melnikov-type condition plus an additional assumption, the negatively and positively asymptotic sets persist under periodic… (More)

We generalize the Newtonian n-body problem to spaces of curvature κ = constant, and study the motion in the 2-dimensional case. For κ > 0, the equations of motion encounter non-collision singularities, which occur when two bodies are antipodal. These singularities of the equations are responsible for the existence of some hybrid solution singularities that… (More)

We study a 2-body problem given by the sum of the Newtonian potential and an anisotropic perturbation that is a homogeneous function of degree −β, β ≥ 2. For β > 2, the sets of initial conditions leading to collisions/ejections and the one leading to escapes/captures have positive measure. For β > 2 and β 6= 3, the flow on the zero-energy manifold is… (More)

- Marshall Hampton, Gareth E. Roberts, Manuele Santoprete
- J. Nonlinear Science
- 2014

We examine in detail the relative equilibria in the planar four-vortex problem where two pairs of vortices have equal strength, that is, Γ1 = Γ2 = 1 and Γ3 = Γ4 = m where m ∈ R−{0} is a parameter. One main result is that for m > 0, the convex configurations all contain a line of symmetry, forming a rhombus or an isosceles trapezoid. The rhombus solutions… (More)

We prove that there is a unique convex non-collinear central configuration of the planar Newtonian four-body problem when two equal masses are located at opposite vertices of a quadrilateral and, at most, only one of the remaining masses is larger than the equal masses. Such central configuration posses a symmetry line and it is a kite shaped quadrilateral.… (More)

In this paper we show that in the n-body problem with harmonic potential one can find a continuum of central configurations for n = 3. Moreover we show a counterexample to an interpretation of Jerry Marsden Generalized Saari’s conjecture. This will help to refine our understanding and formulation of the Generalized Saari’s conjecture, and in turn it might… (More)

In this paper we find a class of new degenerate central configurations and bifurcations in the Newtonian n-body problem. In particular we analyze the Rosette central configurations, namely a coplanar configuration where n particles of mass m1 lie at the vertices of a regular n-gon, n particles of mass m2 lie at the vertices of another n-gon concentric with… (More)

Bi-Hamiltonian structures are of great importance in the theory of integrable Hamiltonian systems. The notion of compatibility of symplectic structures is a key aspect of bi-Hamiltonian systems. Because of this, a few different notions of compatibility have been introduced. In this paper we show that, under some additional assumptions, compatibility in the… (More)