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 \ $(1)\quad \ddot x + g(x) = 0.$\quad Some refinements of known criteria are brought. Moreover, we give necessary and sufficient conditions so that the analytic potential of equation $(1)$ 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\\ $\theta_j(u,\tau), j=1,2,3,4$\ where $\tau=i\pi t, t > 0$. This permits us to prove that\ $\log \frac{\theta_j(u, t)}{\theta_j(0, t)}, j=2,3,4$ and $\log \frac{\theta_1(u, t)}{\pi \theta'_1(0, t)}$ as well as $\frac{\frac{\delta\theta_j}{\delta u}}{\theta_j}$ as functions of $t$ are completely monotonic. We also interested in the quotients $S_j(u,v,t) = \frac{\theta_j(u/2,i\pi t)}{\theta_j(u/2,i\pi t)}$. For fixed $u,v$ such that $0\leq u < v < 1$ we prove that the functions $\frac{(\frac{\delta}{\delta t}S_j)}{S_j}$ for $j=1,4$ as well as the functions $-\frac{(\frac{\delta}{\delta t}S_j)}{S_j}$ for $j=2,3$ are completely monotonic for $t \in ]0,\infty[$.\\ {\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 $T(c)$ (energy-dependent) for potential systems $\ddot x + g(x)=0$ with a center at the origin 0. We had before identified a family of new criteria noted by $(C_n)$ 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 $$(S) \quad \dot x=y,\qquad \dot y = -h(x) - g(x)y - f(x)y^2.$$ We are interested in the period function $T$ around a center 0. A sufficient condition for the isochronicity of (S) at 0 is given. Such a condition is also necessary when $f,g,h$ 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

Isochronicity conditions for some planar polynomial systems II

We study the isochronicity of centers at $O\in \mathbb{R}^2$ for systems $$\dot x=-y+A(x,y),\;\dot y=x+B(x,y),$$ where $A,\;B\in \mathbb{R}[x,y]$, which can be reduced to the Li\'enard type equation. When $deg(A)\leq 4$ and $deg(B) \leq 4$, using the so-called C-algorithm we found $36$ new families of isochronous centers. When the Urabe function $h=0$ we provide an explicit general formula for linearization. This paper is a direct continuation of \cite{BoussaadaChouikhaStrelcyn2010} but can be read independantly