Differential Equations, Dynamical Systems, and an Introduction to Chaos

Citation

Hirsch, M. W., Smale, S., & Devaney, R. L. (2012). Differential Equations, Dynamical Systems, and an Introduction to Chaos (3rd ed.). Academic Press (Elsevier).

Chapter Summary

The book has seventeen chapters. Chapters 1–6 build the linear theory, Chapters 7–10 turn to nonlinear systems and their long-term behaviour, Chapters 11–13 apply this to biology, circuits and mechanics, Chapters 14–16 introduce chaotic dynamics, and Chapter 17 returns to the existence and uniqueness theorem. Twelve of the chapters contain one or more sections titled "Exploration" (seventeen in all in the contents), which pursue a specific model or question beyond the chapter's main line.

Chapter 1: First-Order Equations

The book starts with the simplest equation, then moves to the logistic population model, constant harvesting and bifurcations, and periodic harvesting and periodic solutions. A section on computing the Poincaré map introduces a tool the book uses later, and the exploration studies a two-parameter family.

Chapters 2–4: Planar Linear Systems; Phase Portraits for Planar Systems; Classification of Planar Systems

Chapter 2 starts from second-order equations, turns them into planar systems, reviews the algebra needed, and develops eigenvalues, eigenvectors, solution of linear systems and the linearity principle. Chapter 3 draws the phase portraits that arise from real distinct, complex and repeated eigenvalues, and treats changes of coordinates. Chapter 4 classifies planar systems through the trace–determinant plane and a dynamical notion of classification; its exploration is a three-dimensional parameter space.

Chapters 5–6: Higher-Dimensional Linear Algebra; Higher-Dimensional Linear Systems

Chapter 5 supplies the linear algebra for systems in more than two variables: preliminaries, eigenvalues and eigenvectors, complex eigenvalues, bases and subspaces, repeated eigenvalues, and genericity. Chapter 6 applies it to systems with distinct and repeated eigenvalues, harmonic oscillators, the exponential of a matrix, and nonautonomous linear systems.

Chapters 7–8: Nonlinear Systems; Equilibria in Nonlinear Systems

Chapter 7 defines dynamical systems and states the existence and uniqueness theorem, continuous dependence of solutions and the variational equation; two explorations look at numerical methods and at numerical methods in connection with chaos. Chapter 8 studies equilibria through examples, nonlinear sinks, sources and saddles, stability and bifurcations; the exploration concerns complex vector fields.

Chapters 9–10: Global Nonlinear Techniques; Closed Orbits and Limit Sets

Chapter 9 introduces nullclines, stability of equilibria, gradient systems and Hamiltonian systems, with the pendulum under constant forcing as the exploration. Chapter 10 treats limit sets, local sections and flow boxes, the Poincaré map, monotone sequences in planar systems, the Poincaré–Bendixson theorem and its applications, and finishes with oscillating chemical reactions.

Chapters 11–13: Applications in Biology; Applications in Circuit Theory; Applications in Mechanics

Chapter 11 covers infectious diseases, predator–prey systems and competing species, with explorations on harvesting under competition and on a zombie extension of the SIR model. Chapter 12 starts from the RLC circuit and moves through the Liénard and van der Pol equations to a Hopf bifurcation, with neurodynamics as the exploration. Chapter 13 develops Newton's second law, conservative systems, central force fields, Kepler's first law, the two-body problem and the technique of blowing up a singularity, with explorations on other central force problems, classical limits of quantum systems and the motion of a glider.

Chapters 14–16: The Lorenz System; Discrete Dynamical Systems; Homoclinic Phenomena

Chapter 14 introduces the Lorenz system, its elementary properties, the Lorenz attractor, a geometric model for it and the chaotic attractor; the exploration is the Rössler attractor. Chapter 15 turns to discrete systems: bifurcations, the discrete logistic model, chaos, symbolic dynamics, the shift map and the Cantor middle-thirds set, with cubic chaos and the orbit diagram as explorations. Chapter 16 covers the Shilnikov system, the horseshoe map, the double scroll attractor and homoclinic bifurcations, with the Chua circuit as exploration.

Chapter 17: Existence and Uniqueness Revisited

The last chapter revisits the theorem of Chapter 7 and gives its proof, followed by continuous dependence on initial conditions, extending solutions, nonautonomous systems and differentiability of the flow.

Key Concepts

Chapters 1–6: Linear theory

• Poincaré map: introduced in Chapter 1 for periodically forced first-order equations and used again for closed orbits in Chapter 10.
• Trace–determinant plane: the setting for classifying planar linear systems in Chapter 4.
• Exponential of a matrix: the device in Chapter 6 for solving higher-dimensional linear systems.

Chapters 7–10: Nonlinear systems

• Existence and uniqueness theorem: stated in Chapter 7 and proved in Chapter 17.
• Stability and bifurcations of equilibria: the subject of Chapter 8.
• Gradient and Hamiltonian systems: two classes of systems treated in Chapter 9.
• Poincaré–Bendixson theorem: the planar result on limit sets in Chapter 10.

Chapters 11–17: Applications and chaos

• Hopf bifurcation: the subject of Section 12.4, following the Liénard and van der Pol equations.
• Lorenz attractor: the central example of Chapter 14.
• Symbolic dynamics and the shift map: tools in Chapter 15 for understanding chaos in discrete systems.
• Horseshoe map and homoclinic bifurcation: the subjects of Chapter 16.

Critical Analysis

Strengths

• A clear path from linear to nonlinear. The contents move from planar linear systems, through higher-dimensional linear algebra, to nonlinear equilibria, global techniques and limit sets, so each topic has been prepared by the one before it. Chapter 5 and the preliminaries in Section 2.3 mean that the linear algebra is reviewed inside the book.
• Explorations. Seventeen "Exploration" sections give students a model to investigate on their own, from a two-parameter family in Chapter 1 to the Chua circuit in Chapter 16. The publisher describes the third edition as having updated material and expanded applications; the contents include, for example, a zombie extension of the SIR model (11.5).
• Three application chapters. Biology, circuits and mechanics each receive a full chapter (11–13), which gives the qualitative tools of Chapters 8–10 concrete work to do.
• Theory is not left out. The existence and uniqueness theorem is stated early (Section 7.2) and given a separate proof in Chapter 17 (Section 17.2), so a course can use the theorem first and the proof later.

Limitations

• Not a first course in solution techniques. The contents show no chapters on series solutions, Laplace transforms or partial differential equations, and the contents list second-order equations only in Section 2.1 and harmonic oscillators in Section 6.2; students who need a catalogue of methods for solving equations will want another book alongside it.
• Specialised later chapters. Chapters 13 and 16 (blowing up a singularity, the Shilnikov system, the horseshoe map, homoclinic bifurcations) are the most specialised in the contents, and the publisher gives the book's readership as advanced undergraduates and graduate students.
• Entry level is not sharply defined. The publisher says calculus is required and that linear systems are introduced slowly, yet describes the readership as advanced undergraduates and graduate students; readers should judge for themselves whether they want a first course in differential equations beforehand.

Where It Fits in the Course

This book belongs in the second slot of a differential equations sequence. Boyce and DiPrima's Elementary Differential Equations and Boundary Value Problems is a first course that balances solution methods, qualitative understanding and applications, and Jiří Lebl's Notes on Diffy Qs is a free first course aimed at engineering students; either is the natural thing to read before this book, particularly for the solution methods that Hirsch, Smale and Devaney leave out. Arrowsmith et al.'s Dynamical Systems: Differential Equations, Maps, and Chaotic Behaviour is described as a rigorous treatment of stability, nonlinear dynamics, maps and chaotic behaviour and is a possible next step, and Differential Equations, Stability, and Chaos in Dynamic Economics is the place to go for economic applications of the same ideas.

How to Use This Book

The book suits students of mathematics, science and engineering who have completed calculus and a course in linear algebra and who want the geometric, qualitative side of differential equations. It can also serve self-study readers who have already seen basic ODE techniques elsewhere.

A practical plan:

• Core (Chapters 1–10): read in order. Chapters 2–4 are the foundation for everything planar, and Chapters 7–10 carry the nonlinear theory. Do the explorations in Chapters 1, 4, 8, 9 and 10 for hands-on experience of bifurcations and limit sets.
• Applications (Chapters 11–13): choose according to interest; Chapter 11 on epidemics and populations is the most accessible, and Chapter 13 is best for physics students.
• Chaos (Chapters 14–16): Chapters 14 and 15 (the Lorenz system and discrete dynamical systems) are the core of the chaos material; Chapter 16 is the most specialised and is best left for a second pass.
• Theory (Chapter 17): read after Chapter 7 if the proof of existence and uniqueness is part of the course.

Use a computer algebra system or a plotting tool for the phase portraits and for the numerical explorations in Chapter 7.