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.
NeimarkSacker-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.
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