In this paper we study spiderweb central configurations for the N -body problem, i.e configurations given by N = n × + 1 masses located at the intersection points of concurrent equidistributed half-lines with n circles and a central mass m 0 , under the hypothesis that the masses on the i-th circle are equal to a positive constant m i ; we allow the particular case m 0 = 0. We focus on constructive proofs of the existence of spiderweb central configurations, which allow numerical implementation. Additionally, we prove the uniqueness of such central configurations when ∈ {2, . . . , 9} and arbitrary n and m i ; under the constraint m 1 ≥ m 2 ≥ . . . ≥ m n we also prove uniqueness for ∈ {10, . . . , 18} and n not too large. We also give an algorithm providing a rigorous proof of the existence and local unicity of such central configurations when given as input a choice of n, and m 0 , ..., m n . Finally, our numerical simulations highlight some interesting properties of the mass distribution.
We consider spatially homogeneous Hořava-Lifshitz (HL) models that perturb General Relativity (GR) by a parameter v ∈ (0, 1) such that GR occurs at v = 1/2. We describe the dynamics for the extremal case v = 0, which possess the usual Bianchi hierarchy: type I (Kasner circle of equilibria), type II (heteroclinics that induce the Kasner map) and type VI 0 , VII 0 (further heteroclinics). For type VIII and IX, we prove the existence of periodic orbits which are far from the Mixmaster attractor, and thereby yield a new behaviour which is not described by the BKL picture.
In this paper we study nonlinear autonomous retarded functional differential equations; that is, functional equations where the time derivative may depend on the past values of the variables. When the nonlinearities in such equations are comprised of elementary functions, we give a constructive proof of the existence of an embedding of the original coordinates yielding a polynomial differential equation. This embedding is a topological conjugacy between the semi-flow of the original differential equation and the semi-flow of the auxiliary polynomial differential equation. Further dynamical features are investigated; notably, for an equilibrium or a periodic orbit and its embedded counterpart, the stable and unstable eigenvalues have the same algebraic and geometric multiplicity.
We consider spatially homogeneous Hořava-Lifshitz models that perturb General Relativity (GR) by a parameter v ∈ (0, 1) such that GR occurs at v = 1/2. We describe the dynamics for the extremal case v = 0, which possess the usual Bianchi hierarchy: type I (Kasner circle of equilibria), type II (heteroclinics that induce the Kasner map) and type VI 0 , VII 0 (further heteroclinics). For type VIII and IX, we use a computer-assisted approach to prove the existence of periodic orbits which are far from the Mixmaster attractor. Therefore we obtain a new behaviour which is not described by the BKL picture of bouncing Kasner-like states.
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