334 research outputs found
Isochronicity conditions for some planar polynomial systems II
We study the isochronicity of centers at for systems
where , which
can be reduced to the Li\'enard type equation. When and , using the so-called C-algorithm we found new families of
isochronous centers. When the Urabe function we provide an explicit
general formula for linearization. This paper is a direct continuation of
\cite{BoussaadaChouikhaStrelcyn2010} but can be read independantly
Period function and characterizations of Isochronous potentials
We are interested at first in the study of the monotonicity for the period
function of the conservative equation \ \quad
Some refinements of known criteria are brought. Moreover, we give necessary and
sufficient conditions so that the analytic potential of equation is
isochronous. These conditions which are different from those introduced firstly
by Koukles and Piskounov and thereafter by Urabe appear sometime to be easier
to use. We then apply these results to produce families of isochronous
potentials depending on many parameters, some of them are news. Moreover,
analytic isochronicity requirements of parametrized potentials will also be
consideredComment: 30 page
Complete Monotonicity of classical theta functions and applications
We produce trigonometric expansions for Jacobi theta functions\\
\ where . This permits us to
prove that\ and as well as
as functions of are
completely monotonic. We also interested in the quotients . For fixed such that
we prove that the functions for as well as the functions
for are completely
monotonic for .\\ {\it Key words and phrases} : theta
functions, elliptic functions, complete monotonicity.Comment: 19 page
On the monotonicity criteria of the period function of potential systems
The purpose of this paper is to study various monotonicity conditions of the
period function (energy-dependent) for potential systems with a center at the origin 0. We had before identified a family of new
criteria noted by which are sometimes thinner than those previously
known ({\it Period function and characterizations of Isochronous
potentials}\quad arXiv:1109.4611). This fact will be illustrated by examples.Comment: 9 page
On the period function of Newtonian systems
We study the existence of centers of planar autonomous system of the form
We are interested in the period function around a center 0. A sufficient
condition for the isochronicity of (S) at 0 is given. Such a condition is also
necessary when are analytic functions. In that case a characterization
of isochronous centers of system (S) is given. Some applications will be
derived. In particular, new families of isochronous centers will be describedComment: 16 page
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