A. Raouf Chouikha

Isochronicity conditions for some planar polynomial systems II

Abstract We study the isochronicity of centers at O ∈ R 2 for systems x ˙ = − y + A ( x , y ) , y ˙ = x + B ( x , y ) , where A , B ∈ R [ x , y ] , which can be reduced to the Lienard type equation. Using the so-called C-algorithm we have found 27 new multiparameter isochronous centers.

Isochronicity of analytic systems via Urabe's criterion

A method for studying isochronous oscillations in some systems of ODE reducible to the equation is described. It is applied to obtain the necessary and sufficient conditions for isochronicity of a cubic two-dimensional autonomous system depending on six parameters. For all isochronous systems in this family the Urabe function is explicitly constructed.

On Properties of Elliptic Jacobi Functions and Applications

Abstract In this paper we are interested in developments of the elliptic functions of Jacobi. In particular a trigonometric expansion of the classical theta functions introduced by the author (Algebraic methods and q-special functions, C.R.M. Proceedings and Lectures Notes, A.M.S., vol 22, Providence, 1999, 53–57) permits one to establish a differential system. This system is derived from the heat equation and is satisfied by their coefficients. Several applications may be deduced. Other types of expansions for the Jacobi elliptic functions as well as for the Zeta function are examined.

Series solutions of some anharmonic motion equations

Abstract Consider the class of nonlinear oscillators of the form (1) d 2 u dt 2 +f(u)=ϵg(t), u(0)=a 0 , u′(0)=0, where g(t) is a 2T-periodic function, f is a function only dependent on u and ϵ is a small parameter. We are interested in the periodic solutions with minimal period 2T, when the restoring term f and g satisfy suitable conditions. By using methods of trigonometric series for solving differential equations we prove the existence of a periodic solution of this perturbed Duffing equation. These results develop and extend a previous ones (Appl. Math. Lett. 14 (8) (2001) 363–368).

Note on trigonometric expansions of theta functions

We are interested in properties of coefficients of certain expansions of the classical theta functions. We show that they are solutions of a differential system derived from the heat equation. We plan to explicitly give expressions of these coefficients.

Nonparallelism of the Ricci Tensor of Pseudo-cylindric Metrics

We describe examples of metrics in the conformal class [g] on some conformally flat Riemannian manifolds (M,g]. These metrics have a constant scalar curvature and an harmonic curvature with nonparallel Ricci tensor.We describe examples of metrics in the conformal class [g] on some conformally flat Riemannian manifolds (M,g]. These metrics have a constant scalar curvature and an harmonic curvature with nonparallel Ricci tensor.

Bonnesen-Type Inequalities and Applications

In this paper we discuss about conjectures ennounced in A.R. Chouikha (1999) and produce significant examples to underline the interest of the problem.