This book aims to provide mathematical analyses of nonlinear differential equations, which have proved pivotal to understanding many phenomena in physics, chemistry and biology. Topics of focus are nonlinear oscillations, deterministic chaos, solitons, reaction-diffusion-driven chemical pattern formation, neuron dynamics, autocatalysis and molecular evolution. Included is a discussion of processes from the vantage of reversibility, reflected by conservative classical mechanics, and irreversibility introduced by the dissipative role of diffusion. Each chapter presents the subject matter from the point of one or a few key equations, whose properties and consequences are amplified by approximate analytic solutions that are developed to support graphical display of exact computer solutions.