Abstract
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.
Highlights
Irregular behavior is present in almost every aspect of life, whether natural or artificial
A pioneer of what is known as chaos theory was Lorenz [1] with his study of a small-scale model of the atmosphere
Lorenz observed that the trajectories from the solution of the respective autonomous differential equation system exhibited extreme sensitivity to tiny changes of the initial conditions
Summary
Irregular behavior is present in almost every aspect of life, whether natural or artificial. This refers to those that are between the rigidly predictable and the purely random Systems in this gap are known as chaotic and have some distinguishing properties: nonlinearity, sensitivity to initial conditions, long-time unpredictability. One of the most notable nonlinear systems is the Chua oscillating circuit due to its simplicity, robustness and diverse dynamics. It was elaborated by Leon Chua at Waseda University in Japan in 1983 [3] on the assumption that a chaotic autonomous circuit should exhibit at least two unstable equilibrium points. It is possible to find mechanical and electromechanical systems whose dynamics are governed by the same differential equations of the Chua’s circuit [9].
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