The beginning of the 20th century saw several revolutions in physics, including quantum theory with its strange dualism of waves and particles, relativity with its invocation of nonintuitive geometries, and the intractability of conventional mathematics for dealing with nonlinear systems (exemplified by the three-body problem). While the first two made rapid progress, the third had to await the development of digital computers. In the 1960s, Edward Lorenz discovered the sensitive dependence of such systems on initial conditions while simulating a system of nonlinear differential equations modeling atmospheric convection, leading to an explosion of work in chaos and complex systems, almost all of it heavily dependent on computers.
The fundamental challenge in studying such systems is that their nonlinearity resists analytic solution. Since the work of Lorenz, computers have addressed this problem through numerical integration, allowing scientists to simulate the behavior of equations that they cannot solve.
Roberts explores a new way in which computers can help study such systems. Instead of simulating the equations, he simplifies them to forms that can yield analytic insights. Iterative techniques exist to develop a Taylor series expansion of the solution to complex differential equations, but the algebra necessary to carry out that iteration is formidable. The central insight of this book is that computer algebra packages (such as the freely available Reduce) make such simplification practical. Thus he focuses the power of the computer not on the black-box behavior of the equations that describe a system but on understanding the dominant structure of the equations themselves.
Part 1 develops the basic theme of perturbing an equation, then iterating it to grow a polynomial approximation to the solution. Roberts demonstrates the approach to solve a quadratic equation, and then shows how to apply it to differential equations. Even though the resulting series are often divergent, he develops techniques for constructing an asymptotic approximation to these series and shows how the basic methods can be applied to oscillatory and forced systems.
Part 2 introduces the idea of a center manifold, a low-dimensional region of state space to which trajectories converge. While the math supporting the existence of such manifolds is typically restricted to the neighborhood of the equilibrium, Roberts shows that for many practical problems, this neighborhood can be quite large and sometimes encompass the entire state space. Such a manifold provides a model of the dynamics that is both accurate and of lower dimensionality than the full dynamics, so it is fortunate that iterative techniques supported by computer algebra can construct such manifolds.
Part 3 shows how to apply these methods to the analysis of systems with large spatial extent (such as the flow of fluid through a pipe, or the dispersion of fluid over a large surface). The key insight is that the lateral spatial derivative can be treated as a small parameter, allowing the application of the center manifold techniques from Part 2.
Part 4 develops methods (as always, iterative and exploiting computer algebra) to discover coordinate transformations that separate variables and allow the discovery of normal form equations that are simpler to analyze than the original ones.
Though the focus of the book is on manipulating the equations, ultimately these equations will be simulated, and in most cases this simulation will require discretizing space. Part 5 adapts the methods developed so far to this extension.
Part 6 deals with the problem of oscillations within the center manifold itself and develops methods for recovering the evolution of the amplitude of the oscillations, which is typically much slower than the oscillations themselves.
Part 7 examines techniques for dealing with noise in nonlinear systems, including averaging, coordinate transforms that separate fast from slow dynamics, and techniques of stochastic calculus.
The volume includes numerous worked examples, as well as exercises for each chapter, and provides Reduce code for the various manipulations. The preface advertises a restriction to undergraduate levels of linear algebra, calculus, and differential equations, but the book also draws on concepts from dynamical systems theory that might not be familiar to undergraduates without further reading. The bibliography usefully indicates the pages on which each reference is cited, simplifying the task of retrieving those further concepts.
This book’s detailed exposition, practical orientation, and hands-on approach make it an essential guide for physicists and engineers who need to work with analytically intractable systems. Now, in addition to simulating those systems as black boxes, we can begin to gain insights into their internal behavior.