On the perturbations theory of the Duffing oscillator in a complex domain

La thèse concerne l'étude des cycles limites d'une équation différentielle sur le plan (la deuxième partie du 16ème problème de Hilbert). La notion de "cycle limite" a une grande importance dans la théorie de la stabilité, elle est introduite par Poincaré vers la fin du 19ème siècle et désigne une orbite périodique isolée. Le but de cette thèse est : d'établir l'existence d'une borne supérieure finie, pour le nombre de cycle limites d'une équation quadratique dans le plan. Ce problème est aussi appelé 16ème problème d' Hilbert infinitésimal. Probablement, l'outil le plus fondamental pour l'étude de la stabilité et les bifurcations des orbites périodiques est l'application de Poincaré, défini par Henri Poincaré en 1881. Cependant, la méthode de Melnikov nous donne une excellente procédure pour déterminer le nombre de cycles limites dans une bande continue de cycles qui sont préservés sous perturbation. En effet, le nombre, les positions et les multiplicités des équations différentielles planes perturbées avec une petite perturbation non nulle sont déterminées par le nombre, les positions et les multiplicités des zéros des fonctions génératrices. La fonction de Melnikov est plus précisément, appelé la fonction de Melnikov de premier- ordre. Si cette fonction est identiquement nulle à travers la bande continue de cycles, on calcule ce qu'on appelle " la fonction de Melnikov d'ordre supérieure ". Ensuite, une analyse d'ordre supérieure est nécessaire, ce qui peut être fait par " l'algorithme de Françoise. Les discussions et les calculs présentés dans notre travail sont limités non seulement à la fonction de Melnikov de premier ordre, mais aussi pour les fonctions de Melnikov de deuxième -ordre. Ces outils seront utiles pour résoudre notre problématique. Les activités de recherche menées dans le cadre de cette recherche sont divisées en quatre parties : La première partie de cette thèse, traite les systèmes dynamiques plans et l'existence de cycles limites. Nous souhaitons après résoudre le problème suivant: Calculer la cyclicité de l'oscillateur asymétrique perturbé de Duffing. Dans la deuxième partie, nous sommes intéressés de la cyclicité à l'extérieur de l'anneau périodique de l'oscillateur de Duffing pour une perturbation particulière, puis, nous fournissons un diagramme de bifurcation complet pour le nombre de zéros de la fonction de Melnikov associée dans un domaine complexe approprié en se basant sur le principe de l'argument. Le nombre de cette cyclicité est égal à trois. Dans la troisième partie, nous étudions la cyclicité à l'intérieur ainsi que à l'extérieur de double boucle homocline pour une perturbation cubique arbitraire de l'oscillateur de Duffing en utilisant les mêmes techniques de Iliev et Gavrilov dans le cas d'un Hamiltonien asymétrique de degré quatre. Notre principal résultat est que deux au plus cycle limite peuvent bifurquer de la double homocline. D'autre part, il est représenté, qu'après bifurcation de eight-loop un cycle limite étranger est née, qui ne soit pas contrôlée par un zéro lié par les intégrales Abéliennes, ce cycle supplémentaire est appelé " Alien ".This thesis concerns the study of limit cycles of a differential equation in the plane (The second part of the 16th Hilbert problem). The concept of "limit cycle" has a great importance in the theory of stability; Poincaré introduces this notion at the end of the 19th century and denotes an isolated periodic orbit. The purpose of this thesis: Find an upper bound finite to the number of limit cycles of a quadratic equation in the plane. This problem is so- called the infinitesimal Hilbert 16th problem. Probably, the most basic tool for studying the stability and bifurcations of periodic orbits is the Poincaré, defined by Henri Poincaré in 1881. However, Melnikov's method gives us an excellent method for determining the number of limit Cycles in a continuous band of cycles that are preserved under perturbation. In fact, the number, positions and multiplicities of perturbed planar differential equations for a small nonzero parameters, are determined by the number, positions and multiplicities of the zeros of the generating functions. The Melnikov function is more precisely, called the first-order Melnikov function. If this function is identically equal zero across the continuous band of cycles, one computes the so-called "Higher order Melnikov function". Then, a higher order analysis is necessary which can be done by making use of the so called "the algorithm of Françoise". The discussions and computation presented in this thesis are restricted not only to the first order Melnikov function, but also to the second-order Melnikov functions. These tools will be useful to resolve the question problem. The research activities in the framework of this thesis are divided into four parts: The first part of this thesis, discusses planar dynamical systems and the existence of limit cycles. We wish to solve the following problem: Calculate the cyclicity of the perturbed asymmetric oscillator Duffing. In the second part, we are interested of the cyclicity of the exterior period annulus of the asymmetrically perturbed Duffing oscillator for a particular perturbation, then, we provide a complete bifurcation diagram for the number of zeros of the associated Melnikov function in a suitable complex domain based on the argument principle. The number of this cyclicity is equal to three. In the third part, we study the cyclicity of the interior and exterior eight-loop especially for arbitrary cubic perturbations by using the same techniques of Iliev and Gavrilov in the case of an asymmetric Hamiltonian of degree four. Our main result is that at most two limit cycles can bifurcate from double homoclinic loop. On the other hand, it is appears after bifurcation of eight-loop an "Alien" limit was born, which is not covered by a zero of the related Abelian integrals

Bifurcations with spherical symmetry

Bifurcations from spherically symmetric states can occur in many physical and biological systems. These include the development of a spherical ball of cells into an asymmetrical state and the buckling of a sphere under pressure. They also occur in the evolution of reaction–diffusion systems on a spherical surface and in Rayleigh–Benard convection in a spherical shell. Many of the behaviours of these systems can be explained by their underlying spherical symmetry alone. Using results from the area of mathematics known as equivariant bifurcation theory we can use group theoretical methods both to predict the symmetries of the solutions which are expected to result from bifurcations with symmetry and compute their stability. In this thesis both stationary and Hopf bifurcation with spherical symmetry are discussed. Firstly, using group theoretical techniques, the symmetries of the periodic solutions which can be found at a Hopf bifurcation with spherical symmetry are computed. This computation has been carried out previously but contains some errors which are corrected here. For one particular representation of the group of symmetries of the sphere the stability properties of the bifurcating branches of periodic solutions resulting from the Hopf bifurcation are analysed and a survey is carried out of other periodic and quasiperiodic solutions which can exist. Secondly, symmetry considerations are used to investigate the existence and stability properties of symmetric spiral patterns on the surface of a sphere which result from stationary bifurcations. It is found that in the case of the Swift–Hohenberg equation spiral patterns with one or more arms can exist and be stable on spheres of certain radii. Although one-armed spirals in the Swift–Hohenberg equation are stationary solutions, it is shown that generically one-armed spirals on spheres must drift

Bifurcations with spherical symmetry

Bifurcations from spherically symmetric states can occur in many physical and biological systems. These include the development of a spherical ball of cells into an asymmetrical state and the buckling of a sphere under pressure. They also occur in the evolution of reaction–diffusion systems on a spherical surface and in Rayleigh–Benard convection in a spherical shell. Many of the behaviours of these systems can be explained by their underlying spherical symmetry alone. Using results from the area of mathematics known as equivariant bifurcation theory we can use group theoretical methods both to predict the symmetries of the solutions which are expected to result from bifurcations with symmetry and compute their stability. In this thesis both stationary and Hopf bifurcation with spherical symmetry are discussed. Firstly, using group theoretical techniques, the symmetries of the periodic solutions which can be found at a Hopf bifurcation with spherical symmetry are computed. This computation has been carried out previously but contains some errors which are corrected here. For one particular representation of the group of symmetries of the sphere the stability properties of the bifurcating branches of periodic solutions resulting from the Hopf bifurcation are analysed and a survey is carried out of other periodic and quasiperiodic solutions which can exist. Secondly, symmetry considerations are used to investigate the existence and stability properties of symmetric spiral patterns on the surface of a sphere which result from stationary bifurcations. It is found that in the case of the Swift–Hohenberg equation spiral patterns with one or more arms can exist and be stable on spheres of certain radii. Although one-armed spirals in the Swift–Hohenberg equation are stationary solutions, it is shown that generically one-armed spirals on spheres must drift

Symmetry Method for Limit Cycle Walking of Legged Robots.

Dynamic steady-state walking or running gaits for legged robots correspond to periodic orbits in the dynamic model. The common method for obtaining such periodic orbits is conducting a numerical search for fixed points of a Poincare map. However, as the number of degrees of freedom of the robot grows, such numerical search becomes computationally expensive because in each search trial the dynamic equations need to be integrated. Moreover, the numerical search for periodic orbits is in general sensitive to model errors, and it remains to be seen if the periodic orbit which is the outcome of the search in the domain of the dynamic model corresponds to a periodic gait in the actual robot. To overcome these issues, we have presented the Symmetry Method for Limit Cycle Walking, which relaxes the need to search for periodic orbits, and at the same time, the limit cycles obtained with this method are robust to model errors. Mathematically, we describe the symmetry method in the context of so-called Symmetric Hybrid Systems, whose properties are discussed. In particular, it is shown that a symmetric hybrid system can have an infinite number of periodic orbits that can be identified easily. In addition, it is shown how control strategies need to be selected so that the resulting reduced order system still possesses the properties of a symmetric hybrid system. The method of symmetry for limit cycle walking is successfully tested on a 12-DOF 3D model of the humanoid robot Romeo.PhDApplied and Interdisciplinary MathematicsUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/133356/1/razavi_1.pd