898 research outputs found
A reverse KAM method to estimate unknown mutual inclinations in exoplanetary systems
The inclinations of exoplanets detected via radial velocity method are
essentially unknown. We aim to provide estimations of the ranges of mutual
inclinations that are compatible with the long-term stability of the system.
Focusing on the skeleton of an extrasolar system, i.e., considering only the
two most massive planets, we study the Hamiltonian of the three-body problem
after the reduction of the angular momentum. Such a Hamiltonian is expanded
both in Poincar\'e canonical variables and in the small parameter , which
represents the normalised Angular Momentum Deficit. The value of the mutual
inclination is deduced from and, thanks to the use of interval
arithmetic, we are able to consider open sets of initial conditions instead of
single values. Looking at the convergence radius of the Kolmogorov normal form,
we develop a reverse KAM approach in order to estimate the ranges of mutual
inclinations that are compatible with the long-term stability in a KAM sense.
Our method is successfully applied to the extrasolar systems HD 141399, HD
143761 and HD 40307.Comment: 19 pages, 3 figure
On the convergence of an algorithm constructing the normal form for lower dimensional elliptic tori in planetary systems
We give a constructive proof of the existence of lower dimensional elliptic
tori in nearly integrable Hamiltonian systems. In particular we adapt the
classical Kolmogorov's normalization algorithm to the case of planetary
systems, for which elliptic tori may be used as replacements of elliptic
keplerian orbits in Lagrange-Laplace theory. With this paper we support with
rigorous convergence estimates the semi-analytical work in our previous article
(2011), where an explicit calculation of an invariant torus for a planar model
of the Sun-Jupiter-Saturn-Uranus system has been made. With respect to previous
works on the same subject we exploit the characteristic of Lie series giving a
precise control of all terms generated by our algorithm. This allows us to
slightly relax the non-resonance conditions on the frequencies.Comment: 45 page
An inflationary differential evolution algorithm for space trajectory optimization
In this paper we define a discrete dynamical system that governs the
evolution of a population of agents. From the dynamical system, a variant of
Differential Evolution is derived. It is then demonstrated that, under some
assumptions on the differential mutation strategy and on the local structure of
the objective function, the proposed dynamical system has fixed points towards
which it converges with probability one for an infinite number of generations.
This property is used to derive an algorithm that performs better than standard
Differential Evolution on some space trajectory optimization problems. The
novel algorithm is then extended with a guided restart procedure that further
increases the performance, reducing the probability of stagnation in deceptive
local minima.Comment: IEEE Transactions on Evolutionary Computation 2011. ISSN 1089-778
Improved convergence estimates for the Schr\"oder-Siegel problem
We reconsider the Schr\"oder-Siegel problem of conjugating an analytic map in
in the neighborhood of a fixed point to its linear part, extending
it to the case of dimension . Assuming a condition which is equivalent to
Bruno's one on the eigenvalues of the linear part
we show that the convergence radius of the conjugating transformation
satisfies with
characterizing the eigenvalues , a constant not depending on
and . This improves the previous results for , where the
known proofs give . We also recall that is known to be the optimal
value for .Comment: 21 page
Graph-based algorithms for the efficient solution of a class of optimization problems
In this paper, we address a class of specially structured problems that
include speed planning, for mobile robots and robotic manipulators, and dynamic
programming. We develop two new numerical procedures, that apply to the general
case and to the linear subcase. With numerical experiments, we show that the
proposed algorithms outperform generic commercial solvers.Comment: 27 pages, 9 figures, 1 tabl
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