On the existence of periodic solution of perturbed generalized Li\'enard equations

Under conditions of Levinson-Smith type, we prove the existence of a $\tau$-periodic solution for the perturbed generalized Li\'enard equation u''+\phi(u,u')u'+\psi(u)=\epsilon\omega(\frac{t}{\tau},u,u') with periodic forcing term. Also we deduce sufficient condition for existence of a periodic solution for the equation u''+\sum_{k=0}^{2s+1} p_k(u){u'}^k=\epsilon\omega(\frac{t}{\tau},u,u'). Our method can be applied also to the equation u''+[u^2+(u+u')^2-1]u'+u=\epsilon\omega(\frac{t}{\tau},u,u'). The results obtained are illustrated with numerical examples.Comment: 15 pages, 5 figure

Amplitude Independent Frequency Synchroniser for a Cubic Planar Polynomial System

International audienceThe problem of local linearizability of the planar linear center perturbed by cubic non-linearities in all generalities on the system parameters (14 parameters) is far from being solved. Synchronization problem [18, 5] consists in bringing appropriate modifications on a given system to obtain a desired dynamic. The desired phase portrait along this paper contains a compact region around a singular point at the origin in which lies periodic orbits with the same period (independently from the chosen initial conditions). In this paper, starting from a 5-parameters non isochronous Chouikha cubic system [8], we identify all possible monomial perturbations of degree d ∈ {2, 3} insuring local linearizability of the perturbed system.The necessary conditions are obtained by the Normal Forms method. These conditions are real algebraic equations (multivariate polynomials) in the parameters of the studied ordinary differential system. The efficient algorithm FGb [13] for computing the Gröbner basis is used. For the family studied in this paper, an exhaustive list of possible parameters values insuring local linearizability is established. All the found cases are already known in the literature but the contexts is different since our object is the syn-chronisation not the classification. This paper can be seen as a direct continuation of several new works concerned with the hinting of cubic isochronous centers in particular [8, 12, 15, 10, 6, 2, 3], also it can be considered as an adaptation of a qualitative theory method to a synchronization problem. Mathematics Subject Classification (2010). Primary 34C15, 34C25, 34C37

Isochronicity conditions for some real polynomial systems

This paper focuses on isochronicity of linear center perturbed by a polynomial. Isochronicity of a linear center perturbed by a degree four and degree five polynomials is studied, several new isochronous centers are found. For homogeneous isochronous perturbations, a first integral and a linearizing change of coordinates are presented. Moreover, a family of Abel polynomial systems is also considered. By investigations until degree 10 we prove the existence of a unique isochronous center. These results are established using a computer implementation based on Urabe theorem.Comment: 26 page

Complexity reduction of C-algorithm

The C-Algorithm introduced in [Chouikha2007] is designed to determine isochronous centers for Lienard-type differential systems, in the general real analytic case. However, it has a large complexity that prevents computations, even in the quartic polynomial case. The main result of this paper is an efficient algorithmic implementation of C-Algorithm, called ReCA (Reduced C-Algorithm). Moreover, an adapted version of it is proposed in the rational case. It is called RCA (Rational C-Algorithm) and is widely used in [BardetBoussaadaChouikhaStrelcyn2010] and [BoussaadaChouikhaStrelcyn2010] to find many new examples of isochronous centers for the Li\'enard type equation

Amplitude Independent Frequency Synchroniser for a Cubic Planar Polynomial System

The problem of local linearizability of the planar linear center perturbed by cubic nonlinearities in all generalities on the system parameters (14 parameters) is far from being solved. The synchronization problem (as noted in Pikovsky, A., Rosenblum, M., an

Inverted pendulum stabilization: Characterization of codimension-three triple zero bifurcation via multiple delayed proportional gains

International audienceThe paper considers the problem of stabilization of systems possessing a multiple zero eigenvalue at the origin. The controller that we propose, uses multiple delayed measurements instead of derivative terms. Doing so, we increase the performances of the closed loop in presence of system uncertainties and/or noisy measurements. The problem formulation and the analysis is presented through a classical engineering problem which is the stabilization of an inverted pendulum on a cart moving horizontally. On one hand, we perform a nonlinear analysis of the center dynamics described by a three dimensional system of ordinary differential equations with a codimension-three triple zero bifurcation. On the other hand, we present the complementary stability analysis of the corresponding linear time invariant system with two delays describing the behavior around the equilibrium. The aim of this analysis is to characterize the possible local bifurcations. Finally, the proposed control scheme is numerically illustrated and discussed

Control of Drilling Vibrations: A Time-Delay System-Based Approach

International audienceThe main purpose of this study is the control of both axial and torsional vibrations occurring along a rotary oilwell drilling system. This work completes a previous author's paper [Boussaada I., Mounier H., Niculescu S-I., Cela A. 2012] which presents the description of the qualitative dynamical response of a rotary drilling system with a drag bit, using a model that takes into consideration the axial and the torsional vibration modes of the bit. The studied model, based on the interface bit-rock, contains a couple of wave equations with boundary conditions consisting of the angular speed and the axial speed at the top additionally to the angular and axial acceleration at the bit whose contain a realistic frictional torque. Our analysis is based on the center manifold theorem and normal forms theory whose allow us to simplify the model. By this way we design two control laws allowing to suppress the undesired vibrations guaranteeing a regular drilling process. The obtained results are numerically illustrated

Analysis and control of quadrotor via a Normal Form approach

International audienceThis paper focuses on the analysis and control of some mathematical models representing the dynamics of a quadrotor. By using a normal form approach, the highly coupled parts in the quadrotor system are eliminated, while all possible properties of the original system are not changed. The bifurcations of the system are then analyzed. A two dimensional system is deduced at the origin which can determine the stability and possible local bifurcations of the system. Based on the normal form and indirect method of Lyapunov, we propose a state feedback control method with computational simplicity as well as practical implementation facility. Comparing to a standard PID control, the proposed method has faster response time and less tracking errors especially with wind disturbance

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