Differential Equations, Bifurcations and Chaos

This is the author website for the book Differential Equations, Bifurcations and Chaos, by Paul C. Matthews, published by Springer in September 2025 as part of their Springer Undergraduate Mathematics Series.

The book is aimed at undergraduate students in Mathematics, Physics, Chemistry and Engineering. Because of the broad target audience, it does not include formal mathematical proofs. There are 235 pages, over 60 diagrams, and many worked examples. Each chapter ends with a concise summary of the key points and a selection of exercises.

Chapters

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

• Appendix: Essential Background Mathematics
• Solutions to Even-Numbered Exercises (on-line)

Topics covered include:

• Separation of variables, integrating factor method
• Phase lines for first-order equations, stability and instability of fixed points
• Phase planes for second-order equations, classification of fixed point types
• Examples, including the nonlinear pendulum, dynamics of interacting species, epidemic modelling
• Systems with parameters and the main bifurcations that occur as these parameters vary
• Discrete dynamical systems and their similarities and differences compared with continuous ones
• Chaos in differential equations, such as the Lorenz system, and in discrete systems

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