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.
Adventure with the author as he sails from Koror, Palau among the Micronesian islands to Darwin, Australia on the north coast of the continent. Along the way visit one of the most remote and pristine islands in the Pacific Ocean, meet its two inhabitants (and a few visitors), then continue to sail through pirate infested waters of Indonesia without legal papers. Experience the sea at night, sitting alone on deck, riding the waves and savoring stars. Enjoy the adventure of a first time sailing voyage. More importantly explore the relationship between a father and son, and the long lasting bonds that extend beyond death. Experience the fulfillment of a long held dream, explore a different culture and its people, muse about the twists and turns of life, reflect on the value of family and friends, marvel at the vast creation all around us. Enjoy the voyage, and be encouraged to fulfill your own dreams.
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