Dynamical Systems: Differential Equations, Maps, and Chaotic Behaviour

Citation

Arrowsmith, D. K., & Place, C. M. (1992). Dynamical Systems: Differential Equations, Maps, and Chaotic Behaviour. Chapman & Hall. Chapman & Hall/CRC Mathematics Series, vol. 5.

Chapter Summary

The book has six chapters, a bibliography with hints to the exercises, and an index, in about 330 pages. It replaces the authors' earlier Ordinary Differential Equations (1982). The publisher's synopsis describes a text on the qualitative properties of dynamical systems, covering both differential equations and maps, that relies heavily on examples supported by extensive exercises, hints to solutions and diagrams. Its stated aim is to set the newer topics of chaos and fractals within the standard teaching of ordinary differential equations.

Chapter 1: Introduction

The basic concepts: existence and uniqueness of solutions, the geometrical representation of solutions, and autonomous equations. The emphasis is on phase portraits and on the qualitative view of a system. The chapter ends with exercises.

Chapter 2: Linear Systems

Solution methods and classification of linear systems, phase portraits for the canonical forms, higher-dimensional linear systems, the evolution operator and affine systems. Exercises close the chapter.

Chapter 3: Non-linear Systems in the Plane

Local and global behaviour of planar nonlinear systems, stability of fixed points, first integrals, and the Poincaré–Bendixson theory of limit cycles, together with non-simple fixed points. The index terms include the linearization theorem, separatrices, isoclines and trapping regions.

Chapter 4: Flows on Non-Planar Phase Spaces

Differential equations in phase spaces that are not the plane: hyperbolic and non-hyperbolic fixed points, closed orbits, attracting sets and attractors, Hamilton's equations and Poincaré maps. The Poincaré map links flows to the study of maps developed later.

Chapter 5: Applications I: Planar Phase Spaces

The planar theory applied to models of mechanical oscillators, electrical circuits and coupled oscillators, with affine and non-linear examples including the Van der Pol oscillator.

Chapter 6: Applications II: Non-Planar Phase Spaces, Families of Systems, and Bifurcations

The longest chapter, running from about page 212 to the exercises at page 288. It examines non-planar models and families of systems whose behaviour changes with a parameter, and treats bifurcations, including bifurcations of maps. The index lists area-preserving maps and the flip bifurcation among its terms. According to the synopsis, the book also gives introductions to fractal imaging and cellular automata.

Key Concepts

Chapters 1–2: Phase Portraits and Linear Systems

• Phase portrait: a picture of solution curves in phase space that shows the qualitative behaviour of a system.
• Autonomous equation: an equation whose rate of change depends only on the state, not on time.
• Evolution operator: the operator that carries an initial state forward along a linear system.

Chapters 3–4: Nonlinear Systems

• Stability of fixed points: classification of the behaviour near an equilibrium, including hyperbolic and non-hyperbolic cases.
• Limit cycle and Poincaré–Bendixson theory: isolated periodic orbits of planar systems and the theorem that identifies them.
• Poincaré map: reduction of a flow to a map, connecting differential equations and maps.

Chapters 5–6: Applications and Bifurcations

• Bifurcation: a qualitative change in the dynamics as a parameter varies, as in the saddle-node and flip bifurcations.
• Chaotic behaviour: the irregular dynamics of discrete systems that motivates the title.

Critical Analysis

Strengths

• Examples first. The synopsis says the approach relies heavily on examples, supported by extensive exercises, hints to solutions and diagrams, which suits a subject whose main tool is the phase portrait.
• Differential equations and maps together. The book places chaos and discrete dynamics inside the framework of a standard ordinary differential equations course, using the Poincaré map as the bridge.
• Physical applications. The applications chapters cover mechanical and electrical models, and the synopsis names physics, engineering and geology.
• Support for self-study. The bibliography is followed by hints to the exercises, and each of the first four chapters ends with a set of exercises.

Limitations

• Compact. At about 330 pages and six chapters, the book is a first course rather than a reference; readers who need proofs of the more advanced results or other methods will use another text.
• A product of its date. It was written in 1992 when chaos and fractals were new to the curriculum, as the synopsis says, and later developments are not covered.
• Maps and chaos have no chapters of their own. The title highlights maps and chaotic behaviour, but none of the six chapter titles names them, so a reader who wants discrete dynamics early must look for it inside the chapters on flows and bifurcations.

Where It Fits in the Course

Arrowsmith and Place is the qualitative, dynamical-systems text in this list at the mathematical end. Boyce, DiPrima and Meade's Elementary Differential Equations and Boundary Value Problems and Lebl's free Notes on Diffy Qs introduce the solution methods and make good preparation. Hirsch, Smale and Devaney's Differential Equations, Dynamical Systems, and an Introduction to Chaos treats ODEs from the dynamical-systems point of view and is the closest alternative on the same subject.

Brock and Malliaris's Differential Equations, Stability and Chaos in Dynamic Economics applies related stability methods to economic models, and reading Arrowsmith and Place first gives the mathematical picture that book builds on.

How to Use This Book

The book is suited to students who have completed an introductory course in ordinary differential equations and linear algebra, and who want the qualitative theory with applications in the sciences.

• Core: Chapters 1 to 3, which establish phase portraits and the planar theory.
• Next: Chapter 4 for non-planar flows and Poincaré maps, then the applications in Chapters 5 and 6.
• Practice: do the exercises at the end of each chapter and use the hints at the back only after attempting them. Sketching the phase portraits by hand is the most useful skill to develop.