Chaos and Dynamical Systems
Editor's Note: This article originally appeared in SIAM News (Volume 53, Issue 6, July/August 2020). The original article can be found at https://sinews.siam.org/Details-Page/chaos-and-dynamics-for-all.
While the field of dynamical systems continues to evolve, thus remaining true to its inherent definition, mathematicians’ efforts to utilize these concepts to understand order and disorder are unwavering. David P. Feldman’s new book, Chaos and Dynamical Systems, serves as an introduction to the field and falls somewhere between popular overviews on chaos theory and high-level textbooks on dynamics. His thoughtful approach engages readers at a variety of levels and draws them into the subject’s distinct beauty.
Feldman offers simple, clear explanations that give meaning to his equations and figures and provide deeper insight for those in search of more detail. The book expects readers to be comfortable with algebra and functions, but goes light on calculus. Its readability and careful build-up to more advanced topics avoids an abstract leap that befalls many other texts in this field. Chaos and Dynamical Systems is therefore appropriate for an both undergraduate survey course or a graduate course that covers the broader theme of applied mathematics. Even experts would likely find it to be a nice addition to their libraries.
Feldman addresses modeling in both discrete and continuous dynamical systems, more commonly known as iterated functions and ordinary differential equations. The models are openly motivated as qualitative and intended to be descriptive or empirical, and he evidently chooses each chapter’s classic examples based on their lucidity in standard analysis techniques. Not all simple models behave simply, and Feldman uses chapter three--entitled “Interlude: Mathematical Models and the Newtonian Worldview”--to philosophically discuss why researchers should optimistically continue to believe that the world is understandable. The subsequent two chapters introduce the concept of chaos, and Feldman writes that “Mathematical chaos is exquisitely lawful; a chaotic dynamical system obediently follows a deterministic rule in perpetuity.” He also admits that students are often disappointed by this notion, as they frequently expect something weirder or stranger. Nevertheless, Feldman continues with interesting analyses of the butterfly effect, Lyapunov exponents, and symbolic dynamics, among other topics.
Feldman’s discussion of bifurcations in chapter six is a fascinating journey that “connects the dots” for the reader, so to speak. Using an example that pertains to population dynamics, he demonstrates a bifurcation with discourse, equations, and figures. The end of the chapter explores a broader history of bifurcation theory and its connection to the popular concept of tipping points, complete with numerous suggestions for further reading.
Chapter seven tackles universality in chaos, which Feldman describes as “one of the most amazing results to emerge from the study of dynamical systems.” Appropriately, researchers can employ the logistic function to introduce orbits, state diagrams, and bifurcation diagrams (which provide endless structure for investigation). Feldman then uses Feigenbaum’s constant and period doubling to extend abstract concepts to real-world experimental observations. This process yields a general overview that touches on operators, critical transitions, renormalization, and applications in other physical systems.
Chapter eight turns to the classic Lotka-Volterra predator-prey model and the Rössler equations to introduce phase space and the chaos-related phenomena that are only visible in higher dimensional continuous systems. Feldman then utilizes Poincaré sections and delay coordinates to explore strange attractors in chapter nine. Although he only briefly touches on this incredibly rich subject, the suggestions for further reading in synchronization are excellent.
It is worth noting that Chaos and Dynamical Systems includes a few less traditional topics in dynamics, such as power laws and universality. Feldman concludes with a discussion of the spectrum of opinions on complex systems and emergence. He admits that the fields of complex and dynamical systems are inherently interdisciplinary, and believes that the different approaches keep them vibrant and appealing. Feldman’s enthusiasm for the subject is contagious, and anyone who picks up his book will surely be engaged. After all, how many math books actually describe the discovery of emergent phenomena as exciting and fun?
About the Reviewer
Lora Billings is dean of the College of Science and Mathematics at Montclair State University. Her research focuses on applied deterministic and stochastic dynamical systems that model applications in epidemiology, physics, and ecology.