3 STEPS TO CHAOS .1. EVOLUTION

Linear system theory provides an inadequate characterization of sustained oscillation in nature. In this two-part exposition of oscillation in piecewise-linear dynamical systems, we guide the reader from linear concepts and simple harmonic motion to nonlinear concepts and chaos. By means of three worked examples, we bridge the gap from the familiar parallel RLC network to exotic nonlinear dynamical phenomena in Chua's circuit. Our goal is to stimulate the reader to think deeply about the fundamental nature of oscillation and to develop intuition into the chaos-producing mechanisms of nonlinear dynamics. In order to exhibit chaos, an autonomous circuit consisting of resistors, capacitors, and inductors must contain i) at least one nonlinear element ii) at least one locally active resistor iii) at least three energy-storage elements. Chua's circuit is the simplest electronic circuit that satisfies these criteria. In addition, this remarkable circuit is the only physical system for which the presence of chaos has been proven mathematically. The circuit is readily constructed at low cost using standard electronic components and exhibits a rich variety of bifurcations and chaos. In Part I of this two-part paper, we plot the evolution of our understanding of oscillation from linear concepts and the parallel RLC resonant circuit to piecewise-linear circuits and Chua's circuit. We illustrate by theory, simulation, and laboratory experiment the concepts of equilibria, stability, local and global behavior, bifurcations, and steady-state solutions. In Part II, we study bifurcations and chaos in a robust practical implementation of Chua's circuit.