Chua’s oscillator: an introductory approach to chaos theory

The purpose of this article is to build an introductory approach to chaos theory from the Chua’ oscillator for students of mathematics, physics and engineering. The Chua’ circuit is one of the simplest dynamic systems that produce irregular behavior. Despite its simplicity, this oscillating circuit is a very useful device for studying basic principles of chaos theory. This study begins with the definition of equilibrium points, this is done in a graphic and analytical way. The analysis of the stationary solution of differential equations system allows to show the occurrence of pitchfork bifurcation. The eigenvalues are calculated and the trajectories of the Chua’ oscillator are analyzed, then the occurrence of Hopf bifurcation is demonstrated. Period-doubling cascade are observed using phase portrait and orbit diagram. Finally, sensitivity to initial conditions is studied by calculating Lyapunov exponents.

本文旨在为数学、物理及工程学专业的学生构建一套基于蔡氏振荡器（Chua’ oscillator）的混沌理论入门学习路径。蔡氏电路是能够产生非规则行为的最简动态系统之一。尽管结构简洁，该振荡电路仍是研究混沌理论基本原理的实用实验装置。本研究首先以图解结合解析的方式，对平衡点的定义展开讲解；通过对微分方程组定态解的分析，可展示叉形分岔（pitchfork bifurcation）的发生过程。随后计算特征值并分析蔡氏振荡器的运动轨迹，进而证明霍普夫分岔（Hopf bifurcation）的存在。通过相图（phase portrait）与轨道图（orbit diagram），可观测到倍周期分岔级联（period-doubling cascade）现象。最后，通过计算李雅普诺夫指数（Lyapunov exponents），研究系统对初始条件的敏感性。