Mike R. Jeffrey Knihy

Focusing on models with discontinuities, this book serves mathematicians, scientists, and engineers exploring nonsmooth systems. It features two complementary courses on piecewise smooth flows and maps. The authors progress from established theoretical results to contemporary challenges, covering topics such as stability analysis, bifurcation, singularities, and kneading theory. Rich with examples, the text effectively illustrates the theoretical concepts, making it a valuable resource for those delving into this specialized field.

Focusing on dynamical systems with discontinuities, this volume explores the challenges posed by abrupt changes in system properties, necessitating a non-smooth approach. It reviews current concepts and introduces key methodologies while aiming to stimulate discussion on a significant open problem regarding the fundamental understanding of nonsmooth models. The content is designed to enhance comprehension of how discontinuities affect system behavior and the implications for broader applications in the field.

The dream of mathematical modeling envisions systems evolving in a continuous, deterministic manner. However, continuity is disrupted whenever the 'rules of the game' change, such as shifts in behavioral regimes or physical properties. Discontinuities arise in various contexts, from biological mitosis and seizures to machine malfunctions and earthquakes, as well as individual decisions and economic crashes. These changes lead to a loss of determinacy, and the physical laws governing such transformations are often poorly understood and ill-defined for standard mathematics. Discontinuities allow for the integration of the limits of scientific knowledge into models, enabling detailed analysis while accommodating uncertainty. This approach avoids stochastic modeling, maintaining determinacy as much as possible and focusing on the geometry of potential outcomes when systems break down. The foundations of piecewise-smooth dynamics theory are revitalized through modern nonlinear dynamics and asymptotics. The text includes numerous examples and exercises that guide readers from basic to advanced analytical methods, emphasizing new tools for studying stability and bifurcations. Aimed at scientists and engineers with a foundational understanding of calculus and linear algebra, it serves as a valuable resource for modeling discontinuous systems and encourages readers to create their own innovative models and uncover unknown phenomena.