13,945 research outputs found
Dynamical study while searching equilibrium solutions in N-body problem
The theory of complex dynamics is usually applied to compare the global convergence
properties of different iterative methods, by obtaining the attraction basins for simple
polynomial equations in the complex domain. However, in this work, we use it in quite
another context: the study of a nontrivial nonlinear system that describes the motion of
interacting bodies in celestial mechanics, namely, Newtonian planar circular restricted
four-body problem and its relative equilibrium solutions. These have been investigated
from a dynamical point of view. New properties of the solutions of this system have been
obtained. Practical guidelines for efficient search of relative equilibrium solutions of Nbody
problem have been given.This research has been supported by Ministerio de Ciencia y Tecnologia MTM2014-52016-C2-02.Budzko, D.; Hueso Pagoaga, JL.; MartĂnez Molada, E.; Teruel-Ferragud, C. (2016). Dynamical study while searching equilibrium solutions in N-body problem. Journal of Computational and Applied Mathematics. 297:26-40. https://doi.org/10.1016/j.cam.2015.11.010S264029
MAMA: An Algebraic Map for the Secular Dynamics of Planetesimals in Tight Binary Systems
We present an algebraic map (MAMA) for the dynamical and collisional
evolution of a planetesimal swarm orbiting the main star of a tight binary
system (TBS). The orbital evolution of each planetesimal is dictated by the
secular perturbations of the secondary star and gas drag due to interactions
with a protoplanetary disk. The gas disk is assumed eccentric with a constant
precession rate. Gravitational interactions between the planetesimals are
ignored. All bodies are assumed coplanar. A comparison with full N-body
simulations shows that the map is of the order of 100 times faster, while
preserving all the main characteristics of the full system.
In a second part of the work, we apply MAMA to the \gamma-Cephei, searching
for friendly scenarios that may explain the formation of the giant planet
detected in this system. For low-mass protoplanetary disks, we find that a
low-eccentricity static disk aligned with the binary yields impact velocities
between planetesimals below the disruption threshold. All other scenarios
appear hostile to planetary formation
Stability of Relative Equilibria in the Planar N-Vortex Problem
We study the linear and nonlinear stability of relative equilibria in the
planar N-vortex problem, adapting the approach of Moeckel from the
corresponding problem in celestial mechanics. After establishing some general
theory, a topological approach is taken to show that for the case of positive
circulations, a relative equilibrium is linearly stable if and only if it is a
nondegenerate minimum of the Hamiltonian restricted to a level surface of the
angular impulse (moment of inertia). Using a criterion of Dirichlet's, this
implies that any linearly stable relative equilibrium with positive vorticities
is also nonlinearly stable. Two symmetric families, the rhombus and the
isosceles trapezoid, are analyzed in detail, with stable solutions found in
each case.Comment: 23 pages, 3 figure
Simplest random K-satisfiability problem
We study a simple and exactly solvable model for the generation of random
satisfiability problems. These consist of random boolean constraints
which are to be satisfied simultaneously by logical variables. In
statistical-mechanics language, the considered model can be seen as a diluted
p-spin model at zero temperature. While such problems become extraordinarily
hard to solve by local search methods in a large region of the parameter space,
still at least one solution may be superimposed by construction. The
statistical properties of the model can be studied exactly by the replica
method and each single instance can be analyzed in polynomial time by a simple
global solution method. The geometrical/topological structures responsible for
dynamic and static phase transitions as well as for the onset of computational
complexity in local search method are thoroughly analyzed. Numerical analysis
on very large samples allows for a precise characterization of the critical
scaling behaviour.Comment: 14 pages, 5 figures, to appear in Phys. Rev. E (Feb 2001). v2: minor
errors and references correcte
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