Review of "Discovering Dynamical Systems Through Experiment and Inquiry" by LoFaro and Ford

By Ted Galanthay
Chaos and Dynamical Systems
Discovering Dynamical Systems Through Experiment and Inquiry
Thomas LoFaro and Jeff Ford
CRC Press (Chapman and Hall)
215 pp. (2021)
ISBN: 9780367903947
Reviewed by: Ted Galanthay
School of Humanities and Sciences
Ithaca College
Email: tgalanthay (at)

This is not your typical textbook, but perhaps it should be. Throughout most of its thirteen chapters, the authors entice the reader to learn about discrete dynamical systems through exploration, conjecture, application, and proof. At the companion website, sixty-four graphical applications, whose use is interwoven throughout the text, support students' learning through exploring, recording, and reflecting on their observations. If you aspire to teach in a student-centered classroom, you should consider this textbook, which the authors developed from a collection of classroom activities and demonstrations. The preface provides a brief explanation of how the text can support classroom instruction and seven paragraphs explaining what inquiry-based learning is, how it feels, and why it is important. This text is approachable by students who have taken multivariable calculus and have some experience proving conjectures.

The entire book uses an inquiry-based learning approach and focuses on discrete dynamical systems: what they are and how one analyzes them. Chapter 1 provides an introduction to discrete dynamical systems and defines iterations, orbits, fixed points, and periodic points. Function composition and graphical analysis via cobweb diagrams is presented. The reader is invited to study a population dynamics model by examining assumptions, providing a reasonable function to model reproduction, and exploring the output from simulation.

Chapter 2 defines sequences, subsequences, and Cauchy sequences. Students will build upon ideas from single-variable calculus, namely limits and partial sums, and are introduced to new ideas such as lim inf and lim sup. There is a helpful reminder to the reader on how to interpret absolute values, which is useful in understanding convergence. The chapter finishes with steps to guide the reader to prove that the set of real numbers, with an appropriate metric, is a complete metric space.

Chapters 3 and 4 present linear and nonlinear maps, the implicit function theorem, and fixed point theorems. The reader explores stability of fixed points and periodic points through online apps that are fun to play with before being clearly guided through a four-step process to prove a version of the contraction mapping theorem. Some conjectures may seem prescriptive (e.g., fill in the blanks), but this permits students to create the mathematics necessary to continue the explorations in the intended direction of the authors; an instructor could certainly use these as homework or as the basis for classroom activities for which less scaffolding could be provided.

Chapter 5 introduces tangent and period doubling bifurcations, which the reader can explore in the online apps. This chapter culminates in an analysis of a theorem on the existence of a period doubling bifurcation. Chapter 6 captivates the reader as the reader is led to prove the existence of an infinite number of fixed points and an infinite number of periodic points where previous graphical observations did not reveal these orbits. This exposition leads to the doubling map and the Cantor set in an exciting way. In Chapter 7, the tension builds as the Cantor set is explored and the reader is guided through a sequence of steps to prove properties of this set. The shift map and sequence space on two symbols is introduced in this chapter. Later chapters extend this to N symbols and bi-infinite sequences.

In Chapter 8, the reader is invited, through guided proof steps, to show that the doubling map on [0,1] is conjugate to the shift map and to prove that the logistic map is chaotic. The reader can explore, through an online app, the idea of “sensitive dependence on initial conditions” that goes beyond visualizing separation of nearby points. In Chapter 9, the reader learns the ideas behind computing a bifurcation diagram and explores period doubling bifurcations. This leads to the reader proving a part of Sarkovskii's Theorem, which is introduced in Chapter 10.

Chapter 11 focuses on maps on the plane. After a brief review of essential ideas from linear algebra, the reader is guided to prove results on the linear stability of two-dimensional linear maps. Although a knowledge of linear algebra is not required, students will need to understand what a basis is. The chapter concludes with a stability theorem for nonlinear maps and an explanation for how these results can be applied to determine the stability of periodic points. Chapter 12 introduces the Smale Horseshoe map and develops the proof that the two-sided shift map is chaotic. Chapter 13 provides foundational ideas from topology, builds upon the previous shift maps by expanding to more than two symbols, introduces shifts of finite type, and shows how shift spaces can be represented by graphs.

Overall, this is a wonderful book from which to teach advanced undergraduate students discrete dynamics. Students will need some experience with reading theorem statements and applying proof techniques. The book concentrates on the theory of discrete dynamical systems; thus, there are few examples of applications to other disciplines. Proofs are broken down into manageable steps, though some steps are larger than others. Quite often, the proofs are preceded by conjecture and theorem, which provides an authentic learning experience for the students. Conjectures are often incomplete statements with which students fill in the blanks. Sometimes, these are followed by exercises where students can test their own conjectures and perhaps discover how these need to be reformed. The “explore, conjecture, test, and prove” cycle used throughout much of the text provides an engaging way for students to learn about discrete dynamical systems.

About the Reviewer

Ted Galanthay is an associate professor at the Ithaca College Math Department, in Ithaca, NY. In his research, he builds and analyzes simple mathematical models (equations or systems of equations) of ecological or biological processes. The goal is to discover general principles. Recent motivating questions are “How does evolution affect the optimal movement of organisms?” and “How does natural selection shape animal aggression?”


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