The book deals with continuous piecewise linear differential systems in the plane with three pieces separated by a pair of parallel straight lines. These differential systems are symmetric with respect to the origin of coordinates. This class of systems, driven by concrete applications, is of interest in engineering, in particular in control theory and the design of electric circuits. By studying these particular differential systems, we will introduce the basic tools of the qualitative theory of ordinary differential equations, which allow us to describe the global dynamics of these systems including the dynamics at infinity. The behavior of their solutions, their parametric stability or instability and their bifurcations are described. The book is highly suitable for a first course in the qualitative theory of differential equations or dynamical systems, particularly for engineers, mathematicians, and physicists.

It seems fair to say that while research in linear dynamics is still expanding in both depth and breadth, the foundations have reached a certain stage of maturity. At the same time the basic ideas as well as the applications of the field have a broad appeal also to nonspecialists. It is therefore our aim to make the theory of hypercyclic operators and linear chaos accessible to a wider audience. The book is aimed at advanced undergraduate or beginning graduate students, both as a basis for a lecture course and for self-study. We have strived at a self-contained exposition. Each chapter contains a large number of exercises and ends with a section that gives references and directs the reader to further literature. We have tried to keep the necessary prerequisites for reading this book to a minimum. Since the concept of a hypercyclic operator requires both a topological and a linear structure, the reader is supposed to be familiar with metric spaces (up to the Baire category theorem) and with the basic theory of Hilbert and Banach spaces, as it is often presented in advanced undergraduate courses on analysis. Moreover, since many examples in the theory are given by operators on spaces of holomorphic functions the reader is also expected to have had an introductory course on complex analysis. Additional, more advanced tools that are only needed occasionally will be provided in the two appendices. The book is divided into two parts. Part I presents an introduction to the dynamics of linear operators. Its chapters form a unity and are best studied in that order. In contrast, Part II covers selected topics from linear dynamics. Its chapters are largely independent so that they can be read in an arbitrary order. An occasional cross reference should pose no problem.

This book contains papers about differential algebra, complex variables and orthogonal polynomials. It corresponds to the proceedings of Jairo Charris Seminar 2007-2008

This book is concerning to a Differential Galois (Picard-Vessiot) Theory point of view of the Supersymmetric Quantum Mechanics. The main object is the non-relativistic stationary Schrödinger equation , where are introduced the concepts of Algebraic Spectrum and Hamiltonian Algebrization. Using the Kovacic's Algorithm and the Hamiltonian Algebrization are analyzed Darboux transformations, Crum iterations and supersymmetric quantum mechanics, including their Algebrized Versions from a Galoisian approach. In particular are obtained the ground state, eigenvalues, eigenfunctions, the differential Galois groups and eigenrings of some Schrödinger equations with potentials such as exactly solvable, quasi-exactly solvable and shape invariant potentials. Finally is introduced one methodology to find Algebraically Solvable and Algebraically Quasi- Solvable Potentials. It consists in to apply the Hamiltonian Algebrization, as inverse process, over families of second order linear differential equations integrables in the Picard-Vessiot sense for a set of parameters, in particular, involving orthogonal polynomials and special functions.

This book deals with systems of polynomial autonomous ordinary differential equations in two real variables. The emphasis is mainly qualitative, although attention is also given to more algebraic aspects as a thorough study of the center/focus problem and recent results on integrability. In the last two chapters the performant software tool P4 is introduced. From the start, differential systems are represented by vector fields enabling, in full strength, a dynamical systems approach. All essential notions, including invariant manifolds, normal forms, desingularization of singularities, index theory and limit cycles, are introduced and the main results are proved for smooth systems with the necessary specifications for analytic and polynomial systems.

In this book the problem of station keeping is studied for orbits near libration points in the solar system. The main focus is on orbits near halo ones in the Earth+Moon+Sun system. Taking as starting point the restricted three-body problem, the motion in the full solar system is considered as a perturbation of this simplified model. All the study is done with enough generality to allow easy application to other primary and secondary systems as a simple extension of the analytical and numerical computations.

It is well known that the restricted three-body problem has triangular equilibrium points. These points are linearly stable for values of the mass parameter, m, below Routh's critical value, m1. It is also known that in the spatial case they are nonlinearly stable, not for all the initial conditions in a neighborhood of the equilibrium points L4, L5 but for a set of relatively large measures. This follows from the celebrated Kolmogorov Arnold Moser theorem. In fact there are neighborhoods of computable size for which one obtains "practical stability" in the sense that the massless particle remains close to the equilibrium point for a big time interval (some millions of years, for example). According to the literature, what has been done in the problem follows two approaches: (a) numerical simulations of more or less accurate models of the real solar system; (b) study of periodic or quasi-periodic orbits of some much simpler problem. The concrete questions that are studied in this volume are: (a) Is there some orbit of the real solar system which looks like the periodic orbits of the second approach? (That is, are there orbits performing revolutions around L4 covering eventually a thick strip? Furthermore, it would be good if those orbits turn out to be quasi-periodic. However, there is no guarantee that such orbits exist or will be quasi-periodic). (b) If the orbit of (a) exists and two particles (spacecraft) are put close to it, how do the mutual distance and orientation change with time? As a final conclusion of the work, there is evidence that orbits moving in a somewhat big annulus around L4 and L5 exist, that these orbits have small components out of the plane of the Earth-Moon system, and that they are at most mildly unstable.

This book studies several problems related to the analysis of planned or possible spacecraft missions. It is divided into four chapters. The first chapter is devoted to the computation of quasiperiodic solutions for the motion of a spacecraft near the equilateral points of the Earth-Moon system. The second chapter gives a complete description of the orbits near the collinear point, L1, between the Earth and the Sun in the restricted three-body problem (RTBP) model. In the third chapter, methods are developed to compute the nominal orbit and to design and test the control strategy for the quasiperiodic halo orbits. In the last chapter, the transfer from the Earth to a halo orbit is studied.

The aim of this book is to explain, analyze and compute the kinds of motions that appear in an extended vicinity of the geometrically defined equilateral points of the Earth-Moon system, as a source of possible nominal orbits for future space missions. The methodology developed here is not specific to astrodynamics problems. The techniques are developed in such a way that they can be used to study problems that can be modeled by dynamical systems.

This book introduces the reader to the two main directions of one-dimensional dynamics. The first has its roots in the Sharkovskii theorem, which describes the possible sets of periods of all cycles (periodic orbits) of a continuous map of an interval into itself. The whole theory, which was developed based on this theorem, deals mainly with combinatorial objects, permutations, graphs, etc.; it is called combinatorial dynamics. The second direction has its main objective in measuring the complexity of a system, or the degree of "chaos" present in it; for that the topological entropy is used. The book analyzes the combinatorial dynamics and topological entropy for the continuous maps of either an interval or the circle into itself.