Differential Equations, Bifurcations and Chaos

This book introduces qualitative methods for understanding differential equations, especially when analytical solutions are not possible. Aimed at second-year undergraduate students in mathematics or science, it assumes prior knowledge of calculus, linear algebra, and curve sketching. The book focuses on phase plane methods for second-order differential equations, supported by earlier sections on analytical techniques and phase lines for first-order equations. The later chapters explore bifurcation theory and chaos. Emphasizing application over theory, the book includes diagrams, worked examples, and exercises, with minimal use of formal proofs.

List of contents

Chapter 1. Introduction.- Chapter 2. Analytical Methods for Differential Equations.- Chapter 3. Qualitative Methods for First-Order Differential Equations.- Chapter 4. Second-Order Linear Systems.- Chapter 5. Second-Order Nonlinear Systems.- Chapter 6. Bifurcations.- Chapter 7. Difference Equations.- Chapter 8. Chaos.- Chapter 9. Solutions to Odd-Numbered Exercises.

About the author

Paul C. Matthews was on the faculty of the University of Nottingham for more than two decades. A specialist of dynamical systems and their numerical analysis, he is the author of the bestselling textbook Vector Calculus (Springer, 1998).

Summary

This book introduces qualitative methods for understanding differential equations, especially when analytical solutions are not possible. Aimed at second-year undergraduate students in mathematics or science, it assumes prior knowledge of calculus, linear algebra, and curve sketching. The book focuses on phase plane methods for second-order differential equations, supported by earlier sections on analytical techniques and phase lines for first-order equations. The later chapters explore bifurcation theory and chaos. Emphasizing application over theory, the book includes diagrams, worked examples, and exercises, with minimal use of formal proofs.

Product details

• Authors: Paul C Matthews, Paul C. Matthews
• Publisher: Springer International Publishing
• Languages: English
• Product format: Paperback / Softback
• Release: 30.09.2025
• EAN: 9783031995422
• ISBN: 978-3-031-99542-2
• Illustrations: Approx. 250 p.
• Series: Springer Undergraduate Mathematics Series
• Subjects: Natural sciences, medicine, IT, technology > Mathematics > Analysis; Chaos, Mathematische Physik, Mathematik für Ingenieure, DV-gestützte Biologie/Bioinformatik, Kybernetik und Systemtheorie, Ordinary Differential Equations, Mathematical physics, Differential equations, Mathematical and Computational Biology, Applied Dynamical Systems, Nodes, Lorenz Equations, Bifurcations, Transcritical bifurcation, Phase line, Undergraduate textbook on chaos, Center manifold, Unstable fixed point, Spirals, Saddles, Saddle-node bifurcation, Pitchfork bifurcation, Nullclines, Phase plane, Hopf bifurcation, Applications of differential equations to modeling, Non-autonomous equations