Modeling with Nonsmooth Dynamics (Frontiers in Applied Dynamical Systems: Reviews and Tutorials)

Description

This volume looks at the study of dynamical systems with discontinuities. Discontinuities arise when systems are subject to switches, decisions, or other abrupt changes in their underlying properties that require a ‘non-smooth’ definition. A review of current ideas and introduction to key methods is given, with a view to opening discussion of a major open problem in our fundamental understanding of what nonsmooth models are.  What does a nonsmooth model represent: an approximation, a toy model, a sophisticated qualitative capturing of empirical law, or a mere abstraction? Tackling this question means confronting rarely discussed indeterminacies and ambiguities in how we define, simulate, and solve nonsmooth models. The author illustrates these with simple examples based on genetic regulation and investment games, and proposes precise mathematical tools to tackle them. The volume is aimed at students and researchers who have some experience of dynamical systems, whether as a modelling tool or studying theoretically. Pointing to a range of theoretical and applied literature, the author introduces the key ideas needed to tackle nonsmooth models, but also shows the gaps in understanding that all researchers should be bearing in mind. Mike Jeffrey is a researcher and lecturer at the University of Bristol with a background in mathematical physics, specializing in dynamics, singularities, and asymptotics. Preface Contents 1 Mathematics for a Nonsmooth World 1.1 What Are Discontinuities Hiding? 1.2 Control Switching 1.3 Sticky Genes 1.4 The Plan 2 1930–2010: Nonsmooth Dynamics' Linear Age 3 Discontinuities to Model Missing Knowledge 4 Three Experiments 4.1 Filippov's Convexity Paradox 4.2 On or Off Genes 4.3 Jittery Investments 5 Layers and Implementations 5.1 Linear Switching 5.2 Nonlinear Switching 5.3 Hidden Terms: The `Ghosts' of Switching 6 Ideal and Non-ideal Sliding 6.1 Sliding Perspective I: The Piecewise-Smooth System 6.2 Sliding Perspective II: Hybrid Implementations 6.3 Sliding Perspective III: Smoothed Implementations 7 The Three Experiments Revisited 7.1 Filippov's Paradox Revisited 7.2 Genes Revisited 7.3 Investments Revisited 8 Further Curiosities of Hidden Dynamics 8.1 The Phenomenon of Jitter 8.2 Hidden Oscillations and Chaos 9 Closing Remarks: Open Challenges A Nonsmooth Models as Asymptotic Series A.1 The General Expression A.2 The Example of the Error Function B Simple Examples of Hulls, Canopies, and Indexing C Deriving the Hidden Terms in Lemma 6.2 References Index