Ordinary Differential Equations
Authors
Description
The book offers a modern and in-depth introduction to ordinary differential equations, combining qualitative and numerical analysis of the solutions, following Poincaré’s vision for the field, more than a century ago. Considering the extraordinary development of dynamical systems since then, the book presents the central topics that every young mathematician, whether in the pure or applied area, should know. The approach is always dynamic: the motivating questions, the style of exposition and the proofing techniques are all guided by this perspective.
The text is structured in six cycles. The first addresses the fundamental questions of the existence and uniqueness of solutions. The second presents the basic tools, theoretical and practical, necessary to deal with concrete problems. The third deals with linear theory, both autonomous and non-autonomous. Lyapunov’s theory of stability constitutes the fourth cycle. The fifth examines local theory, including the Grobman–Hartman theorem and the stable variety theorem. Finally, the sixth cycle explores global issues in the broader context of differential equations in manifolds, culminating in the Poincaré–Hopf index theorem.
The work can be used both in undergraduate or graduate courses, or even for individual study. As a prerequisite, it is desirable that the reader has basic knowledge of general topology, linear algebra, and analysis at the undergraduate level. At the end of each chapter, there is a computational experiment, a varied list of exercises and detailed historical, biographical and bibliographical notes, which help the reader to better understand how these ideas developed over time.
Target audience
Higher education
Name: Ordinary Differential Equations
Author(s): Marcelo Viana e José M. Espinar
Pages: 548
Publication: IMPA, 2025
ISBN: 978-85-244-0513-6
Edition: First
1 Introduction
1.1 Differential equations and their solutions
1.2 Qualitative theory of differential equations
1.3 Numerical analysis of differential equations
1.4 Experiment: population dynamics
1.5 Exercises
1.6 Notes
2 Local solutions
2.1 Picard’s Existence and Oneness Theorem
2.1.1 Fixed points of contractions
2.1.2 Proof of the Theorem 2.4
2.1.3 Definition range estimates
2.2 Peano’s Existence Theorem
2.2.1 Approximation by differentiable functions
2.2.2 Equicontinuity
2.2.3 Completion of the demo
2.3 Continuous Dependency Theorem
2.3.1 Continuous Parameter Dependency
2.3.2 Continuous Dependency Theorem
2.4 Differentiable Dependency Theorem
2.4.1 Hadamard’s lemma
2.4.2 Differentiability C1
2.4.3 Proof of the Theorem 2.23
2.4.4 Differentiable Dependency Theorem
2.5 Generalizations
2.5.1 Higher-Order Equations
2.5.2 Partial differential equations 2.6 Experiment: Picard’s method
2.7 Exercises
2.8 Notes
3 Maximum solutions
3.1 Existence and uniqueness
3.2 Behavior at the extremes
3.3 Globally Lipschitzian equations
3.3.1 Gronwall lemma
3.3.2 Proof of the Theorem 3.9
3.4 Continuous Dependency Theorem (global)
3.4.1 Continuous Parameter Dependency
3.4.2 Continuous Dependency Theorem
3.5 Differentiable Dependency Theorem (global)
3.6 Experiment: continuation of solutions
3.7 Exercises
3.8 Notes
4 Numerical integration
4.1 Euler’s method
4.1.1 Formulation
4.1.2 Error Estimates
4.2 Runge–Kutta methods
4.2.1 Heun’s method
4.2.2 The Runge–Kutta family
4.2.3 Upper dimensions
4.3 Convergence of unistep methods
4.3.1 Convergence and consistency
4.3.2 Stability
4.4 Adams’ methods
4.4.1 Adams–Bashforth methods
4.4.2 Adams–Moulton methods
4.5 Convergence of multi-step methods
4.5.1 Convergence, consistency and stability
4.5.2 Accuracy of Adams’ methods
4.6 Stiffness
4.6.1 What is Rigidity?
4.6.2 Implicit methods
4.7 Experiment: contour lines
4.8 Exercises
4.9 Notes
5 Autonomous equations
5.1 Flow of an autonomous equation
5.1.1 Regular, periodic and stationary trajectories
5.1.2 Complete equations
5.1.3 Non-autonomous or higher-order equations
5.2 Tubular Flow Theorem
5.3 Poincaré transformations
5.3.1 Existence and differentiability
5.3.2 Periodic trajectories
5.4 Conjugation and equivalence of flows
5.5 Poincaré’s Recurrence Theorem
5.6 Experiment: Electrical circuits
5.7 Exercises
5.8 Notes
6 Autonomous linear equations
6.1 Exponential of a linear application
6.2 Exponential calculation
6.2.1 Nilpotent applications
6.2.2 Diagonalizable applications
6.2.3 Canonical Jordan Form – real case
6.2.4 Almost diagonalizable applications
6.2.5 Canonical form of Jordan – complex case
6.3 The two-dimensional case
6.3.1 A has two distinct real eigenvalues
6.3.2 A has a single real eigenvalue
6.3.3 A has no real eigenvalues
6.4 Differentiable conjugation of linear flows
6.5 Topological classification of hyperbolic flows
6.5.1 Hyperbolic linear attractors and repulsors
6.5.2 Topological classification theorem
6.6 Experiment: Aerodynamic instability
6.7 Exercises
6.8 Notes
7 Non-autonomous linear equations
7.1 Space of solutions of the homogeneous equation
7.2 Fundamental solutions of the homogeneous equation
7.3 Liouville–Ostrogradsky formula
7.3.1 Application to Nonlinear Autonomous Equations
7.3.2 Application to higher-order linear equations – Wronskian
7.4 Space of solutions of the inhomogeneous equation
7.5 Floquet’s theorem
7.5.1 Linear Application Logarithms
7.6 Experiment: Resonance
7.7 Exercises
7.8 Notes
8 Lyapunov stability
8.1 Autonomous equations: linear stability
8.1.1 Linear equations
8.1.2 Quasi-linear equations
8.2 Autonomous equations: Lyapunov functions
8.2.1 Lyapunov Stability Theorem
8.2.2 Invariant Set Theorem
8.3 Lyapunov analysis of non-autonomous equations
8.3.1. Uniform stability
8.3.2 Lyapunov functions
8.3.3 Additional Comments
8.4 Linear stability and Lyapunov exponents
8.4.1 Linear equations
8.4.2 Quasi-linear equations
8.4.3 Exponents of Lyapunov
8.5 Experiment: Lyapunov’s greatest exponent
8.6 Exercises
8.7 Notes
9 Grobman–Hartman theorem
9.1 Hyperbolic stationary points
9.1.1 Simple stationary points
9.1.2 Hyperbolic Stationary Points
9.2 Grobman–Hartman theorem for flows
9.3 Proof of the Grobman–Hartman Theorem
9.3.1 Globalization of dynamics
9.3.2 Discreet time
9.3.3 Completion of the Demo
9.4 Grobman–Hartman theorem for diffeomorphisms
9.5 Differentiable conjugation
9.6 Experiment: ballistic method
9.7 Exercises
9.8 Notes
10 Stable Variety Theorem
10.1 Local stable and unstable varieties
10.2 Stable Variety Theorem
10.3 Proof of the Stable Variety Theorem
10.3.1 Chart Transformation
10.3.2 Differentiability C1
10.3.3 Ck Differentiability
10.3.4 Conclusion
10.4 Hyperbolic periodic trajectories
10.5 Experiment: Planetary Systems
10.6 Exercises
10.7 Notes
11 Vector Fields on Surfaces
11.1 Sets –limit and !–limit
11.2 Poincaré–Bendixson theorem
11.2.1 Consequences of the Closed Curve Theorem
11.2.2 Completion of the Demo
11.2.3 Van der Pol’s equation
11.3 Boundary sets of flows on surfaces
11.3.1 Differential equations in manifolds
11.3.2 Poincaré–Bendixson theorem in the sphere
11.3.3 Minimal sets
11.4 Mayer’s theorem on conservative flows
11.4.1 Invariant cross-sectional measure
11.4.2 Domains of the Poincaré transformations
11.4.3 Stability and periodic components
11.4.4 Recurrence and Minimal Components
11.4.5 Completion of the Proof of Theorem 11.21
11.5 Structural Stability Comments
11.6 Experiment: Lorenz attractor
11.7 Exercises
11.8 Notes
12 Poincaré–Hopf theorem
12.1 Index of a stationary point
12.1.1 Number of Turns Around a Point
12.1.2 Vector fields in the plane
12.1.3 Vector Fields on Surfaces
12.1.4 Index in higher dimensions
12.2 Euler’s characteristic
12.2.1 Polyhedra
12.2.2 Surfaces
12.3 Indexes and curvature
12.4 Proof of the theorem
12.5 Comments on Mayer’s Theorem
12.6 Experiment: oxygen–ozone cycle
12.7 Exercises
12.8 Notes
A Metric spaces and varieties
A.1 Metric spaces and sequences
A.2 Continuous applications
A.3 Differentiable varieties and applications
A.4 Tangent space and derived application
A.5 Cotangent space and differential forms
A.6 Transversality
A.7 Riemannian varieties
A.8 Euler’s characteristic
A.9 Curvature and connection forms
A.10 Notes
Bibliography
Author Index
Index