## About the book.

I should say first that this is an academic book, not an informal read. It introduces some exciting new things we have learnt about the mathematics needed to describe a world full of discontinuous things, but is written for the academic (student, lecturer, or ageing researcher), needing to develop or analyse dynamical models with discontinuities.

It is often said that Newton's science gave us a way to describe the world and predict its motions, at least in theory, forever. Write down the equations of planetary motion and you can predict how they will spin around the heavens into eternity. But even neglecting the exotic small-scale world of quantum physics, this could hardly be further from the truth. There are numerous situations, from rocking bar stools to piles of sand to ladders propped up against walls, that evade such simple predictive understanding.

So the dream of mathematical modeling, which should have described a world of weather systems and human interactions and cars and planes, all evolving in a continuous, deterministic, predictable way, is a myth. One important reason is that this dream relies on such systems changing in a nice smooth way, and instead we find a world full of events that constantly change the rules, destroying smoothness. Physical properties like density or opacity change across the boundaries of solid bodies. Cells divide, nervous systems seize. Machine parts stick and rattle, the earth rips and quakes. People and governments and companies make decisions that entirely change the landscape we live in, and the equations that describe it.

*Hidden Dynamics* is the behaviour that Nature allows to slip through the cracks as a system jumps from one behaviour to another.

When such *discontinuities* occur, the determinacy we were promised in a post-Newtonian world is inevitably lost. And with it, the mathematics that is taught in schools and universities, and that computer infrastructures take as standard, break down entirely. To describe the world around discontinuities requires a different approach to the mathematics.

Building on nearly a century of pioneering nods in a new direction, this book seeks to provide a substantially new framework to describe discontinuities when they arise in dynamically changing systems.

Often at discontinuities, familiar physical or biological rules break down, and peering into the discontinuity is like peering into a bottomless crevasse. In this book I try to show how discontinuities can be used to make the incompleteness of our scientific knowledge a part of the model, to analyse a system with detail and rigour, yet still leave room for uncertainty (though without assuming uncertainty requires randomness).

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.

My intention is to provide solutions to the exercises here in the near future, so watch this space, and feel free to bug me for them here.

Thanks for visiting Nonsmoothland!