Lessons from Ordinary Differential Equations

Authors

Jorge Sotomayor

Description

This book is based on the courses on Ordinary Differential Equations given by the author in 1971 and 1973 at the Institute of Pure and Applied Mathematics, for graduate students
oriented towards the Master’s Degree in Mathematics.

We develop here the Theory of Ordinary Differential Equations, that is, the study of the general properties of the functions that are solutions of this type of equations, based on the
broad hypotheses about the functions that define them, using the resources of Classical Mathematical Analysis and Linear Algebra, without necessarily resorting to the particular form
of the equations.

The material presented does not differ essentially from that developed in various classical or modern treatises, especially in a foreign language. We register here our
recognition for the influence received from them, and especially mention Coddington and Levinson (1955), Hartman (1964) and Pontryagin (1962).

The book is divided into four basically self-sufficient parts: Foundations, Linear Equations, Qualitative Theory and Structural Stability. A table of interdependence between chapters allows direct shortcuts to various topics, without necessarily following the order in which they are presented in the text. That is, achieved at the cost of a certain repetition of the fundamentals of the Theory, in several versions, which, in our view, complement each other to give a broader view of the available methods. The content of the chapters is dictated by an inevitable choice within the vast universe of Differential Equations. We believe, however, to have addressed the elements of most of the subjects that arose or were systematized from the great movement of foundation and expansion that the Theory experienced in the nineteenth century and that even today, in various forms, are the object of research or used in the applications of the Ordinary Differential Equations.

The text presents more material than could reasonably be covered in the course of a school term, and in this case a careful choice of topics for a first reading is indispensable. The central text is complemented with several appendices, in which we develop important aspects of the Theory but technically more elaborate or different in focus, whose introduction in the text would break the continuity of elementary ideas and methods that predominate in it.

Target audience

Higher education

Name: Lessons from Ordinary Differential Equations

Author(s): e Jorge Sotomayor

Pages: 424

Publication: IMPA, 2025

ISBN: 978-85-244-0597-6

Edition: Second

Foreword

Introduction

I Pleas

1 Existence and uniqueness of solutions
1.1 Preliminaries
1.2 Cauchy’s problem
1.3 Examples
1.4 Picard’s and Peano’s theorems
1.5 Maximum solutions
1.6 Higher-order differential systems and equations
Exercises

2 Dependence of solutions on initial conditions and parameters
2.1 Preliminaries
2.2 Continuity
2.3 Differentiability
Exercises

II Linear Equations

3 Linear differential equations
3.1 Preliminaries
3.2 General properties
3.3 Linear equations with constant coefficients
3.4 Simple two-dimensional systems
3.5 Conjugation of linear systems
3.6 Classification of hyperbolic linear systems
3.7 Complex linear systems
3.8 Mechanical and electrical oscillations
Exercises

4 Elements of Sturm–Liouville Theory and Boundary Problems
4.1 Sturm’s Theorems
4.2 Sturm–Liouville problems
4.3 Existence of eigenvalues
4.4 The Vibrating Rope Problem
4.5 Expansion in series of autofunctions
Appendix: The Spectral Theorem
Exercises

5 Linear equations in the complex field
5.1 Singular points of a linear system
5.2 Simple singular points
5.3 Formal solutions in simple singular points
5.4 Fundamental matrices at a simple singular point
5.5 The equation of order n
5.6 Second-order Fuchsian equations
5.7 The Frobenius method
5.8 The Hypergeometric equation
5.9 The Bessel equation
5.10 Bessel functions and the oscillating membrane equation
Exercises

III Qualitative Theory

6 Qualitative theory of ODEs: general aspects
6.1 Vector fields and flows
6.2 Differentiability of vector field flows
6.3 Phase Portrait of a Vector Field
6.4 Equivalence and conjugation of vector fields
6.5 Local structure of hyperbolic singular points
6.6 Local structure of periodic orbits
6.6.1 The Poincaré transformation
6.6.2 Limit cycles in the plane
6.6.3 Derived from the Poincaré transformation
6.7 Linear flows in the torus
Exercises

7 Poincaré–Bendixson theorem
7.1 Limit and !limit sets of an orbit
7.2 The Poincaré–Bendixson Theorem
7.3 Singular points within a periodic orbit
7.4 The equations of Liénard and van der Pol
Exercises

8 Stability in Liapounov’s sense
8.1 Liapounov stability
8.2 The Liapounov Criterion
8.3 Chetaev’s theorem
Exercises

9 Local structure of singular points and hyperbolic periodic orbits
9.1 Preliminaries
9.2 Hartman’s theorem for diffeomorphisms and hyperbolic periodic orbits
9.3 Hartman’s theorem in Banach spaces
9.4 Hartman’s theorem for vector fields and flows
9.5 Hartman’s theorem: Local case for diffeomorphisms
9.6 Hartman’s Theorem: Local Case for Vector Fields
9.7 Invariant varieties
Appendix: Differentiability of Hyperbolic Point Invariant Manifolds
Exercises

10 Poincaré–Bendixson theory on surfaces
10.1 Number of rotation
10.2 Schwartz’s theorem
Exercises

IV Structural Stability

11 Structural stability
11.1 Preliminary concepts
11.2 The r(M) class ${displaystyle textstyle sum }$ of structurally stable fields
11.3 Structurally stable field aperture and density
11.3.1 Aperture of ${displaystyle textstyle sum }$ r (M)
11.3.2 Topological equivalence of structurally stable fields
11.3.3 Density of ${displaystyle textstyle sum }$ r (M)
11.3.4 Characterization of structurally stable fields
11.3.5 Conclusion
Exercises

12 Forks
12.1 Introduction
12.2 First-order structural stability and bifurcations
12.3 Statement of Key Results
12.3.1 Proof of Theorems 12.11 and 12.13
12.3.2 Laminar structure of ${displaystyle textstyle sum }$ r 1 and accumulation
12.3.3 Cross-Paths A ${displaystyle textstyle sum }$ r 1 and Structural Stability
Exercises

Mathematical Analysis

Bibliography

Author Index

Glossary of Notations

Index