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We consider the Hill four-body problem where three oblate, massive bodies form a relative equilibrium triangular configuration, and the fourth, infinitesimal body orbits in a neighborhood of the smallest of the three massive bodies. We…
We study the stability regions and families of periodic orbits of two planets locked in a co-orbital configuration. We consider different ratios of planetary masses and orbital eccentricities, also we assume that both planets share the same…
The existence of stable bound orbits of test particles is one of the most characteristic properties in black hole spacetimes. In higher-dimensional black holes, due to the dimensionality of gravity, there is no stable bound orbit balanced…
In this paper we show that in the planar circular restricted three body problem there are either infinitely many symmetric consecutive collision orbits or at least one periodic symmetric consecutive collision orbit for all energies below…
Consider four point particles with equal masses in the euclidean space, subject to the following symmetry constraint: at each instant they are symmetric with respect to the dihedral group $D_2$, that is the group generated by two rotations…
We consider the classical three-body problem with an arbitrary pair potential which depends on the inter-body distance. A general three-body configuration is set by three "radial" and three angular variables, which determine the shape and…
We present a method for proving the existence of symmetric periodic, heteroclinic or homoclinic orbits in dynamical systems with the reversing symmetry. As an application we show that the Planar Restricted Circular Three Body Problem…
The flow of a $k$-cooperative system maps the set of vectors with up to~$(k-1)$ sign variations to itself. Strongly $2$-cooperative systems satisfy a strong \Poincare-Bendixson property: any bounded solution that evolves in a compact set…
We numerically investigate the dynamics of orbits in 3D circumbinary phase-space as a function of binary eccentricity and mass fraction. We find that inclined circumbinary orbits in the elliptically-restricted three-body problem display a…
We study the dynamics of a piecewise map defined on the set of three pairwise nonparallel, nonconcurrent lines in $\mathbb{R}^2$. The geometric map of study may be analogized to the billiard map with a different reflection rule so that each…
We apply the symmetry reduction method of Roberts to numerically analyze the linear stability of a one-parameter family of symmetric periodic orbits with regularizable simultaneous binary collisions in the planar pairwise symmetric…
In this contribution, the optimal stabilization problem of periodic orbits is studied via invariant manifold theory and symplectic geometry. The stable manifold theory for the optimal point stabilization case is generalized to the case of…
We study the secular gravitational dynamics of quadruple systems consisting of a hierarchical triple system orbited by a fourth body. These systems can be decomposed into three binary systems with increasing semimajor axes, binaries A, B…
We generalize the Newtonian n-body problem to spaces of curvature k=constant, and study the motion in the 2-dimensional case. For k>0, the equations of motion encounter non-collision singularities, which occur when two bodies are antipodal.…
We analyze the role of local geometry in the spin and orbital interaction in transition metal compounds with orbital degeneracy. We stress that the tendency observed for the most studied case (transition metals in O$_6$ octahedra with one…
Despite the huge number of research into the three-body problem in physics and mathematics, the study of this problem still remains relevant both from the point of view of its broad application and taking into account its fundamental…
This paper considers a class of delay differential equations with unimodal feedback and describes the structure of certain unstable sets of stationary points and periodic orbits. These unstable sets consist of heteroclinic connections from…
We study the existence of asymptotically stable periodic trajectories induced by reset feedback. The analysis is developed for a planar system. Casting the problem into the hybrid setting, we show that a periodic orbit arises from the…
In three space dimensions, when a physical system possesses spherical symmetry, the dynamical equations automatically lead to the Legendre and the associated Legendre equations, with the respective orthogonal polynomials as their standard…
Currently, the fifteen new periodic solutions of Newtonian three-body problem with equal mass were reported by \v{S}uvakov and Dmitra\v{s}inovi\'{c} (PRL, 2013) [1]. However, using a reliable numerical approach (namely the Clean Numerical…