Periodic Differential Equations in the Plane

A Topological Perspective

Rafael Ortega

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English

De Gruyter
06 May 2019

Differential calculus & equations; Geometry; Topology

Series: De Gruyter Series in Nonlinear Analysis & Applications

"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."

By: Rafael Ortega
Imprint: De Gruyter
Country of Publication: Germany
Volume: 29
Dimensions: Height: 240mm, Width: 170mm, Spine: 13mm
Weight: 464g
ISBN: 9783110550405
ISBN 10: 3110550407
Series: De Gruyter Series in Nonlinear Analysis & Applications
Pages: 195
Publication Date: 06 May 2019
Audience: Professional and scholarly , Undergraduate , Undergraduate
Format: Hardback
Publisher's Status: Active

Rafael Ortega, University of Granada, Spain.