The dream of mathematical modeling is of systems evolving in a
continuous, deterministic, predictable way. Unfortunately continuity is
lost whenever the `rules of the game' change, whether a change of
behavioural regime, or a change of physical properties. From biological
mitosis to seizures. From rattling machine parts to earthquakes. From
individual decisions to economic crashes.
Where discontinuities occur, determinacy is inevitably lost. Typically
the physical laws of such change are poorly understood, and too
ill-defined for standard mathematics. Discontinuities offer a way to
make the bounds of scientific knowledge a part of the model, to analyse
a system with detail and rigour, yet still leave room for uncertainty.
This is done without recourse to stochastic modeling, instead retaining
determinacy as far as possible, and focussing on the geometry of the
many outcomes that become possible when it breaks down.
In this book the foundations of `piecewise-smooth dynamics' theory are
rejuvenated, given new life through the lens of modern nonlinear
dynamics and asymptotics. Numerous examples and exercises lead the
reader through from basic to advanced analytical methods, particularly
new tools for studying stability and bifurcations. The book is aimed at
scientists and engineers from any background with a basic grounding in
calculus and linear algebra. It seeks to provide an invaluable resource
for modeling discontinuous systems, but also to empower the reader to
develop their own novel models and discover as yet unknown phenomena.
