Periodic Differential Equations In The Plane
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Periodic Differential Equations in the Plane
Author | : Rafael Ortega |
Publsiher | : Walter de Gruyter GmbH & Co KG |
Total Pages | : 195 |
Release | : 2019-05-06 |
Genre | : Mathematics |
ISBN | : 9783110551167 |
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Periodic differential equations appear in many contexts such as in the theory of nonlinear oscillators, in celestial mechanics, or in population dynamics with seasonal effects. The most traditional approach to study these equations is based on the introduction of small parameters, but the search of nonlocal results leads to the application of several topological tools. Examples are fixed point theorems, degree theory, or bifurcation theory. These well-known methods are valid for equations of arbitrary dimension and they are mainly employed to prove the existence of periodic solutions. Following the approach initiated by Massera, this book presents some more delicate techniques whose validity is restricted to two dimensions. These typically produce additional dynamical information such as the instability of periodic solutions, the convergence of all solutions to periodic solutions, or connections between the number of harmonic and subharmonic solutions. The qualitative study of periodic planar equations leads naturally to a class of discrete dynamical systems generated by homeomorphisms or embeddings of the plane. To study these maps, Brouwer introduced the notion of a translation arc, somehow mimicking the notion of an orbit in continuous dynamical systems. The study of the properties of these translation arcs is full of intuition and often leads to "non-rigorous proofs". In the book, complete proofs following ideas developed by Brown are presented and the final conclusion is the Arc Translation Lemma, a counterpart of the Poincaré–Bendixson theorem for discrete dynamical systems. Applications to differential equations and discussions on the topology of the plane are the two themes that alternate throughout the five chapters of the book.
Almost Periodic Differential Equations
Author | : A.M. Fink |
Publsiher | : Springer |
Total Pages | : 345 |
Release | : 2006-11-15 |
Genre | : Mathematics |
ISBN | : 9783540383079 |
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Playing Around Resonance
Author | : Alessandro Fonda |
Publsiher | : Birkhäuser |
Total Pages | : 309 |
Release | : 2016-11-11 |
Genre | : Mathematics |
ISBN | : 9783319470900 |
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This book provides an up-to-date description of the methods needed to face the existence of solutions to some nonlinear boundary value problems. All important and interesting aspects of the theory of periodic solutions of ordinary differential equations related to the physical and mathematical question of resonance are treated. The author has chosen as a model example the periodic problem for a second order scalar differential equation. In a paedagogical style the author takes the reader step by step from the basics to the most advanced existence results in the field.
Ordinary Differential Equations
Author | : Nicolas Rouche,J. Mawhin |
Publsiher | : Pitman Advanced Publishing Program |
Total Pages | : 280 |
Release | : 1980 |
Genre | : Differential equations |
ISBN | : UCAL:B4405941 |
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Good,No Highlights,No Markup,all pages are intact, Slight Shelfwear,may have the corners slightly dented, may have slight color changes/slightly damaged spine.
The Restricted 3 Body Problem Plane Periodic Orbits
Author | : Alexander D. Bruno |
Publsiher | : Walter de Gruyter |
Total Pages | : 377 |
Release | : 2011-05-03 |
Genre | : Mathematics |
ISBN | : 9783110901733 |
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The aim of the series is to present new and important developments in pure and applied mathematics. Well established in the community over two decades, it offers a large library of mathematics including several important classics. The volumes supply thorough and detailed expositions of the methods and ideas essential to the topics in question. In addition, they convey their relationships to other parts of mathematics. The series is addressed to advanced readers wishing to thoroughly study the topic. Editorial Board Lev Birbrair, Universidade Federal do Ceará, Fortaleza, Brasil Victor P. Maslov, Russian Academy of Sciences, Moscow, Russia Walter D. Neumann, Columbia University, New York, USA Markus J. Pflaum, University of Colorado, Boulder, USA Dierk Schleicher, Jacobs University, Bremen, Germany
Periodic Differential Equations
Author | : F. M. Arscott |
Publsiher | : Elsevier |
Total Pages | : 295 |
Release | : 2014-05-16 |
Genre | : Mathematics |
ISBN | : 9781483164885 |
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Periodic Differential Equations: An Introduction to Mathieu, Lamé, and Allied Functions covers the fundamental problems and techniques of solution of periodic differential equations. This book is composed of 10 chapters that present important equations and the special functions they generate, ranging from Mathieu's equation to the intractable ellipsoidal wave equation. This book starts with a survey of the main problems related to the formation of periodic differential equations. The subsequent chapters deal with the general theory of Mathieu's equation, Mathieu functions of integral order, and the principles of asymptotic expansions. These topics are followed by discussions of the stable and unstable solutions of Mathieu's general equation; general properties and characteristic exponent of Hill's equation; and the general nature and solutions of the spheroidal wave equation. The concluding chapters explore the polynomials, orthogonality properties, and integral relations of Lamé's equation. These chapters also describe the wave functions and solutions of the ellipsoidal wave equation. This book will prove useful to pure and applied mathematicians and functional analysis.
Ordinary Differential Equations
Author | : David A. Sanchez |
Publsiher | : American Mathematical Soc. |
Total Pages | : 132 |
Release | : 2002-12-31 |
Genre | : Mathematics |
ISBN | : 9781470458348 |
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For the instructor or student confronting an introductory course in ordinary differential equations there is a need for a brief guide to the key concepts in the subject. Important topics like stability, resonance, existence of periodic solutions, and the essential role of continuation of solutions are often engulfed in a sea of exercises in integration, linear algebra theory, computer programming and an overdose of series expansions. This book is intended as that guide. It is more conceptual than definitive and more light-hearted than pedagogic. It covers key topics and theoretical underpinnings that are necessary for the study of rich topics like nonlinear equations or stability theory. The [Author]; has included a great many illuminating examples and discussions that uncover the conceptual heart of the matter.
Non Linear Differential Equations
Author | : G. Sansone,R. Conti |
Publsiher | : Elsevier |
Total Pages | : 550 |
Release | : 2016-06-06 |
Genre | : Mathematics |
ISBN | : 9781483135960 |
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International Series of Monographs in Pure and Applied Mathematics, Volume 67: Non-Linear Differential Equations, Revised Edition focuses on the analysis of the phase portrait of two-dimensional autonomous systems; qualitative methods used in finding periodic solutions in periodic systems; and study of asymptotic properties. The book first discusses general theorems about solutions of differential systems. Periodic solutions, autonomous systems, and integral curves are explained. The text explains the singularities of Briot-Bouquet theory. The selection takes a look at plane autonomous systems. Topics include limiting sets, plane cycles, isolated singular points, index, and the torus as phase space. The text also examines autonomous plane systems with perturbations and autonomous and non-autonomous systems with one degree of freedom. The book also tackles linear systems. Reducible systems, periodic solutions, and linear periodic systems are considered. The book is a vital source of information for readers interested in applied mathematics.