44 research outputs found
Bounded analytic maps, Wall fractions and ABC flow
In this work we study the qualitative properties of real analytic bounded
maps defined in the infinite complex strip. The main tool is approximation by
continued g-fractions of Wall. As an application, the ABC flow system is
considered which is essential to the origin of the solar magnetic field.Comment: 14 pages, submitted. arXiv admin note: text overlap with
arXiv:1210.480
On the convergence of continued fractions at Runckel's points and the Ramanujan conjecture
We consider the limit periodic continued fractions of Stieltjes appearing as Shur--Wall -fraction
representations of certain analytic self maps of the unit disc , . We precise the convergence behavior and prove the general
convergence [2, p. 564 ] of (1) at the Runckel's points of the singular line
It is shown that in some cases the convergence holds in the
classical sense. As a result a counterexample to the Ramanujan conjecture [1,
p. 38-39] stating the divergence of a certain class of limit periodic continued
fractions is constructed.Comment: 8 page
On the existence of polynomial first integrals of quadratic homogeneous systems of ordinary differential equations
We consider systems of ordinary differential equations with quadratic
homogeneous right hand side. We give a new simple proof of a result already
obtained in [8,10] which gives the necessary conditions for the existence of
polynomial first integrals. The necessary conditions for the existence of a
polynomial symmetry field are given. It is proved that an arbitrary homogeneous
first integral of a given degree is a linear combination of a fixed set of
polynomials.Comment: 9 pages. to appear in . Phys. A: Math. Gen. 33, 200
Continued g-fractions and geometry of bounded analytic maps
In this work we study qualitative properties of real analytic bounded maps.
The main tool is approximation of real valued functions analytic in rectangular
domains of the complex plane by continued g-fractions of Wall. As an
application, the Sundman-Poincar\'e method in the Newtonian three-body problem
is revisited and applications to collision detection problem are considered.Comment: 16 pages, 6 figures, submitte
The meromorphic non-integrability of the three-body problem
We study the planar three-body problem and prove the absence of a complete
set of complex meromorphic first integrals in a neighborhood of the Lagrangian
solution. We use the Ziglin's method and study the monodromy group of the
corresponding normal variational equations.Comment: 17 pages, submitted to Crelle's Journa
On some exceptional cases in the integrability of the three-body problem
We consider the Newtonian planar three--body problem with positive masses
, , . We prove that it does not have an additional first
integral meromorphic in the complex neighborhood of the parabolic Lagrangian
orbit besides three exceptional cases ,
, where the linearized equations are shown to be partially
integrable. This result completes the non-integrability analysis of the
three-body problem started in our previous papers and based of the
Morales-Ramis-Ziglin approach.Comment: 7 page
The meromorphic non-integrability of the planar three-body problem
We study the planar three-body problem and prove the absence of a complete
set of complex meromorphic first integrals in a neighborhood of the Lagrangian
solution.Comment: 4 pages, Frenc
On some collinear configurations in the planar three-body problem
We study the planar Newtonian three-body problem and analyse the
configurations in which the three bodies or their velocities are collinear. The
existence of such configurations, also called generalised syzygies, was
previously investigated by the author in [4] for bounded solutions. In this
paper we generalise our result to the case of negative energy and provide a
more simple proof. We also study periodic solutions admitting a particular
geometric rigidity and show that they always suffer syzygies i.e. collinear in
positions configurations.Comment: submitte