by David John Warwick Simpson (University of British Columbia, Canada)

Table of Contents (89k)
Preface (133k)
Chapter 1: Fundamentals of Piecewise-Smooth, Continuous Systems (1,102k)

Real-world systems that involve some non-smooth change are often well-modeled by piecewise-smooth systems. However there still remain many gaps in the mathematical theory of such systems. This doctoral thesis presents new results regarding bifurcations of piecewise-smooth, continuous, autonomous systems of ordinary differential equations and maps. Various codimension-two, discontinuity induced bifurcations are unfolded in a rigorous manner. Several of these unfoldings are applied to a mathematical model of the growth of Saccharomyces cerevisiae (a common yeast). The nature of resonance near border-collision bifurcations is described; in particular, the curious geometry of resonance tongues in piecewise-smooth continuous maps is explained in detail.

Neimark–Sacker-like border-collision bifurcations are both numerically and theoretically investigated. A comprehensive background section is conveniently provided for those with little or no experience in piecewise-smooth systems.

• Fundamentals of Piecewise-Smooth, Continuous Systems
• Discontinuous Bifurcations in Planar Systems
• Codimension-Two, Discontinuous Bifurcations
• The Growth of Saccharomyces cerevisiae
• Codimension-Two, Border-Collision Bifurcations
• Periodic Solutions and Resonance Tongues
• Neimark–Sacker-Like Bifurcations