
by A. J. Roberts
Society for Industrial and Applied Mathematics
760 pp. (2015)
ISBN: 9781611973556

Department of Philosophy
Cognitive Sciences Program
University of Central Florida
Email:luis.favela (at) ucf.edu 

The word “complex” in “complexsystems theory” is not equivalent to “complicated.” Phenomena investigated in the science of complexsystems theory are usually conceptualized in terms of features such as nonlinearity, emergence, and the interactions of many physical processes at multiple spatial and temporal scales. Research in the physical sciences—and ever more in the life and social sciences (see, e.g., Favela et al., 2016; Guastello et al., 2011; and Riley & Van Orden, 2005)—require theories and methods to investigate these kinds of systems. Given that complex systems usually have a temporal dimension, the mathematical tools of dynamicalsystems theory are fundamental to their understanding (Amon, 2015; Ball et al., 2013). It is no wonder that there is increasing interest in modern dynamicalsystems theory.
A. J. Roberts begins Model Emergent Dynamics in Complex Systems with the relatively modest goal of demonstrating how to “derive simple dynamical equations that model complex physical interactions,” and to do so “with as elementary mathematics as possible” (p. ix). Roberts attempts to demonstrate that modeling methodologies such as averaging (p. 574), multiple scales (p. 316), and singular perturbations (p. 12) have shortcomings. The purpose of highlighting these limitations is to provide support for the more controversial theme of the book “that coordinate transformations and center manifolds provide a powerfully enhanced and unified view of a swath of other complex system modeling methodologies” (p. ix). Taken together, this unified approach is intended to provide a powerful but practical approach to modeling—and thereby understanding—emergent dynamics in complex systems.
Roberts begins by discussing how modern dynamicalsystems theory brings together algebra and geometry. Algebra provides tools for quantifying systems by capturing the relevant parameters, and geometry provides tools for capturing the qualitative features of systems in a state space (pp. 226–227). Although algebraic equations—such as some linear differential equations of toy examples—can be solved analytically, realistic systems usually cannot be (p. 5). This is where the need for geometry comes in. Because numerical solutions to differential equations may not provide accurate predictions of a system’s dynamics, viewing the geometry of solutions in state space may reveal much more information, albeit of a qualitative nature.
Roberts begins with a simple twodimensional linear ODE as a toy system, and he remarks that even though it has an analytical solution, such a simple method will not work for differential equations of realistic systems. The reason is that differential equations of realistic systems are nonlinear and capture emergent dynamics that typically develop at multiple spatial and temporal scales (pp. xi, 257, 320, and 604). The analysis of such systems can be very difficult. A linear toy system can be solved easily, but complex systems typically require a perturbative approach. Perturbation methods are implemented iteratively. This leads to another important feature of Roberts’ approach: computational algebra.
In addition to the theoretical tools, Roberts discusses the importance of computational algebra as part of his unified toolbox for modeling emergent dynamics in complex systems. Software such as REDUCE allow one to implement the perturbational and iterative approach. In addition to the impracticality of conducting countless calculations by hand, computational algebra is implemented for another important reason: the iterative approach provides simple and reliable algorithms (p. 174). Though the beginnings of model selection can be highly subjective, once initial approximations are made, algorithms via programs such as REDUCE can facilitate the iterative process and reveal the quantitative and qualitative solutions (cf. Jaccard, 2013; Morrison, 2015).
Roberts’ prescription is to invoke centermanifold theory and follow these steps (p. 154):
 “Embed the physical problem in a useful family of cognate problems.”
 Equilibria: Approximate global dynamics by anchoring an equilibrium point in state space.
 Linearization: Linearize dynamics to identify range of modes of the system.
 Theorems: “Invoke existence and emergence theorems, as far as possible, to assure us of the relevance of the modeling.”
 Construction: Construct approximations, usually via computational algebra, to the center/slow manifold.
 Convergence: Find “evidence that the approximations are sufficiently accurate at parameter values of interest.”
 Interpretation: “Identify how many components in the model relate to the original physical process.”
 Regularization: Improve the model; that is, reduce the error.
By following these steps, Roberts appeals to the algebraic and geometrical strengths of dynamicalsystems theory and incorporates the best aspects of other modeling approaches (e.g., averaging, multiple scales, and singular perturbations). With this approach, Roberts discusses a range of phenomena, such as bifurcations, fluid dynamics (e.g., coffee spills on p. 222), oscillators, and stochastic nonautonomous systems.
This is certainly an impressive book. Nonetheless, it has weaknesses. First, although it may seem obvious that a book with this title would be challenging, it is particularly demanding when viewed in light of one of Roberts’ goals of demonstrating how to “derive simple dynamical equations that model complex physical interactions ... with as elementary mathematics as possible” (p. ix). Without a solid background in mathematics and data nalysis software packages, readers are in for a steep uphill battle. Given that complexity science provides theories and methods that are increasingly applied to interdisciplinary and multidisciplinary projects, such a required background in mathematics and software could narrow the scope of his audience. This raises the second weakness of the book: To get the most out of the lessons and exercises, one needs to use the book in conjunction with the REDUCE software package. Although free, REDUCE utilizes different notation and commands than other software packages, such as MATLAB or Mathematica, with which many readers may be more familiar. The third and main weakness of this book is its organization. For example, although the iterative approach is discussed from the beginning of the book (e.g., p. 7), it is not until chapter five (p. 169) that the term is defined. Such jumping around occurs far too often and makes the ideas and lessons more challenging to follow. It could be argued that many of these concepts need not be explained earlier in the book or at all because readers should have a particular background before engaging with this text. However, if it is reasonable to assume knowledge of dynamicalsystems theory, why then is “state space” defined at all, and why is it defined in chapter six (p. 226) after being discussed repeatedly throughout the text and starting in chapter one (p. 6)? These various issues notwithstanding, this book has far more strengths than weaknesses.
The first strength of this book is the language. Although these ideas are very complex (no pun intended), Roberts is able to present many of them clearly, sometimes via simple, yet very helpful, analogies—for example, thinking about coordinate transformations in terms of movement on two car lanes (p. 4). Second, Roberts has bolstered the case for dynamical systems theory and computational methods for investigating and understanding complex systems and related phenomena like nonlinearity, emergence, and multiscale interactions. He has accomplished this by clarifying some of the methods necessary for the study of complex systems. Third, and related to the second strength, Roberts has made an admirable attempt at synthesizing various theories and methods. It is open to debate if Roberts’ reliance on a centermanifold approach is superior to others, such as the method of multiple scales (p. 160). Still, there is no doubt that his prescription for modeling (p. 154) makes theoretical and methodological progress in our ability to model complex systems. Moreover, this prescription utilizes much of what is best about other methods while discarding their weaknesses.
While this book is likely to be very challenging for the mathematically and computationally uninitiated, Roberts has made an exceptional attempt at synthesizing a wide range of theories and methods into a coherent approach to modeling emergent dynamics in complex systems.
References
Amon, M. J. (2015). Review of Complexity science: The Warwick Master’s Course by R. Ball, V. Kolokoltsov, and R. S. MacKay. Dynamical Systems Magazine, October 2015.
Ball, R., Kolokoltsov, V., & MacKay, R. S. (Eds.). (2013). Complexity Science: The Warwick Master’s Course. New York, NY: Cambridge University Press.
Favela, L. H., Coey, C. A., Griff, E. R., & Richardson, M. J. (2016). Fractal analysis reveals subclasses of neurons and suggests an explanation of their spontaneous activity. Neuroscience Letters, 626, 5458.
Guastello, S. J., Koopmans, M., & Pincus, D. (Eds.). (2011). Chaos and Complexity in Psychology: The Theory of Nonlinear Dynamical Systems. Cambridge, MA: Cambridge University Press.
Jaccard, J. (2013). Theory construction, model building, and model selection. In T. D. Little (Ed.), The Oxford Handbook of Quantitative Methods: Volume 1: Foundations (pp. 82104). New York, NY: Oxford University Press.
Morrison, M. (2015). Reconstructing Reality: Models, Mathematics, and Simulations. New York, NY: Oxford University Press.
Riley, M. A., & Van Orden, G. C. (Eds.). (2005). Tutorials in Contemporary Nonlinear Methods for Behavioral Sciences. National Science Foundation, United States.