An Introduction to Piecewise Smooth Dynamics

A focused introduction to nonsmooth dynamics

An Introduction to Piecewise Smooth Dynamics brings together the essentials of a field where ordinary smooth behavior gives way to jumps, switches, and thresholds. This is not a general calculus text or a broad survey of dynamical systems; it is a concentrated graduate-level course on the geometry and behavior of systems that are smooth almost everywhere, but not everywhere.

The book is written with mathematicians, scientists, and engineers in mind, especially readers who work with models that include discontinuities. Its two authors split the subject into the two settings that matter most in practice: continuous-time piecewise-smooth flows and discrete-time piecewise-smooth maps.

Two complementary chapters, two ways of thinking

The first chapter, by Mike R. Jeffrey, introduces piecewise-smooth flows. It moves from the basic history and applications of discontinuous systems into key ideas such as inclusions and combinations, crossing and sliding at switching surfaces, switching layers, codimension-r structures, boundary equilibrium bifurcations, and discontinuity-induced phenomena. Later sections look at determinacy breaking, hidden attractors, hidden bifurcations, and other topics that point toward current research questions.

The second chapter, by Paul Glendinning, focuses on piecewise-smooth maps. Here the course shifts to interval maps, Lorenz-like maps, gluing bifurcations, plane maps, periodic orbits, resonance, robust chaos, and two-dimensional attractors. The result is a compact but wide-ranging path through the discrete-time side of nonsmooth dynamics.

Why this book stands out

• It is grounded in an actual advanced course, so the material is organized pedagogically rather than as a dense reference work.
• It connects classical dynamical-systems ideas with the newer phenomena that arise when smoothness fails at thresholds.
• It balances theory with the kinds of models used in applications, making the subject feel concrete rather than purely abstract.

Who will find it useful

This ebook will be a strong fit for graduate students, researchers, and technical readers who need an introduction to nonsmooth or piecewise-smooth dynamics. If you work in differential equations, applied mathematics, control, or mathematical modeling, the structure of the book makes it easy to move from foundational ideas into more specialized topics.

For readers building a digital library in mathematics, this is a concise and carefully framed introduction to a subject that keeps showing up wherever real systems cross thresholds, switch states, or change rules midstream.