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Les principaux résultats obtenus dans ce travail concernent l’existence et l’unicité des solutions de différents types de l’équation de Liénard forcée et des résultats de dépendance pour les solutions S-asymptotiquement w-périodiques d’équations d’évolution. Pour réaliser notre objectif, nous utilisons des outils d’analyse fonctionnelle non linéaire et des résultats sur des équations linéaire.No English summary available

OSCILLATIONS DANS DES ÉQUATIONS DE LIÉNARD ET DES ÉQUATIONS D'ÉVOLUTION SEMI-LINÉAIRES

In this work, near an equilibrium, we study the existence, uniqueness and regular dependence of almost periodic (a.p.), almost automorphic (a.a.), asymptotically a.p., asymptotically a.a., pseudo a.p., pseudo a.a., weighed pseudo a.p., weighed pseudo a.a. solutions of Liénard differential equations in the form x''(t) + f(x(t), p). x'(t) + g(x(t), p) = ep(t), (1) where the term ep possesses a similar nature, and where p is a parameter in Banach space. We use a theorem of implicit functions around an equilibrium. We also study two special cases of (1), that are x''(t) + f1(x(t)). x'(t) + g1(x(t))= e(t), x''(t) + f2(x(t), q). x'(t) + g2(x(t), q) = e(t). We establish also, a new result of differentiable dependence on the S-asymptotically almost-periodic solution of a Cauchy problem x'(t)=A(t) x(t)+f(t, x(t),u(t) ) x(0) = ζ. The dependence concerns the initial conditions ζ and control term u. We apply this result to parabolic evolution equation with periodic coefficients in time.Dans ce travail, on étudier, au voisinage d'un point d'équilibre, l'existence et l'unicité et la dépendance régulière des solutions presque-périodique (p.p.), présqu'automorphe (p.a.), asymptotiquement p.p., asymptotiquement p.a., pseudo p.p., pseudo p.a., pseudo p.p. avec poids, pseudo p.a. avec poids de la famille d'équations de Liénard forcée suivantes x''(t) + f(x(t), p). x'(t) + g(x(t), p) = ep(t), (1) où le terme ep est de la même nature que la solution, et p est un paramètre dans un espace de Banach. On utilise le théorème des fonctions implicites au voisinage de l'équilibre. On étudier aussi deux cas particuliers de la famille (1) qui sont x''(t) + f1(x(t)). x'(t) + g1(x(t))= e(t), x''(t) + f2(x(t), q). x'(t) + g2(x(t), q) = e(t). On établit aussi un nouveau résultat sur la dépendance différentielle des solutions S-asymptotiquement presque-périodique du problème de Cauchy x'(t)=A(t) x(t)+f(t, x(t),u(t) ) x(0) = ζ , par rapport à la condition initial et le contrôle u. On applique cet résultat sur une équation parabolique avec coefficients périodique par rapport au temps

Small almost periodic and almost automorphic oscillations in forced Liénard equations

International audienceWe consider forced Liénard differential equations in the form x''(t) + f(x(t),p)x'(t) + g(x(t),p) = e(p,t), where p is a parameter in a Banach space. Starting from the zero solution when p=0, we prove the existence of almost periodic (respectively almost automorphic) solutions x(p,.) when e(p,.) is almost periodic (respectively almost automorphic), when p belongs to a neighborhood of 0. We also consider the equations x''(t) + f(x(t))x'(t) + g(x(t)) = e(t) and x''(t) + f(x(t),q)x'(t) + g(x(t),p) = e(t) which are special cases of the previous equations

Architectures of a digital modulator for high-speed data transmission in telecommunication system

Digital systems are more than necessary in our daily life. This last decade, the world is dealing with a remarkable technological revolution, especially, in the field of miniaturization of electronic devices such as modems and embedded systems. This work is focused on challenges of developing Quadrature amplitude modulation (QAM) digital modulators. In this context, this paper proposes several methods, simpler and efficient compared to the existing methods. The obtained results indicate that the proposed technique is effective in terms of data rotation calculation, particularly, for the sinusoidal signal generation in the QAM digital modulators. Through this study, an FPGA-based system with a maximum frequency of 550MHz is developed. The objective is to identify the optimal configuration facilitating data transmission in radio software. The obtained results were compared to recent works, and have showed a significant improvement in terms of computation time, especially in case of increasing the frequency of the transmission system at a very high rate.Keyword: Pipeline cordic, QAM, maximum frequency, throughpu

Several kinds of oscillations in forced Lienard equations

Near an equilibrium we study the existence of asymptotically a.p. (almost periodic), asymptotically a.a. (almost automorphic), pseudo a.p., pseudo a.a., weighed pseudo a. p. and weighed pseudo a. a. solutions of Lienard differential equations in the form x ''(t) + f(x(t),p).x '(t) + g(x(t),p) = e(p)(t), where the forcing term possesses a similar nature, and where p is a parameter in a Banach space. We use a perturbation method around an equilibrium. We also study two special cases of the previous family of equations that are x ''(t) + f(x(t)).x '(t) + g(x(t)) = e(t) and x ''(t) + f(x(t),q).x '(t) + g(x(t),q) = e(t)