Topological methods in nonlinear analysis

Topological Methods for Nonlinear Oscillations

Introduction Periodic phenomena play a pervasive role in natural and in man-made systems. They are exhibited, for example, in simple mathematical models of the solar system and in the observed circadian rhythms by which basic biological functions are regulated. Electronic devices producing stable periodic signals underlie both the electrification of the world and wireless communications. My int...

Topological Methods for Nonlinear Differential Equations — a First Introduction

Topological Methods for Motion Data Analysis

We present a new method for analyzing imagebased motion data with topological methods for flow fields. A video sensor captures motion of subjects and yields discrete samples of velocity vectors in the image domain. We show how to construct a smooth, continuous, globally defined bidirectional flow field which evaluates determined vectors at given samples. We then apply a topological analysis of ...

Methods of Stability Analysis in Nonlinear Mechanics*

We review our recent work on methods to study stability in nonlinear mechanics, especially for the problems of particle accelerators, and compare our ideas to those of other authors. We emphasize methods that (a) show promise as practical design tools, (b) are effective when the nonlinearity is large, and (c) have a strong theoretical basis. TABLE OF CONTENTS

Fixed Point Methods in Nonlinear Analysis

In this paper we present a selection of fixed point theorems with applications in nonlinear analysis. We begin with the Banach fixed point theorem, which we use to prove the inverse and implicit mapping theorems and the Picard-Lindelöf theorem for Banach spaces. We then prove in succession the fixed point theorems of Brouwer, Schauder, and Schaeffer, after which we conclude with two example app...