In this tutorial paper we present one of the simplest autonomous differential equations capable of generating chaotic behavior. Some of the fundamental routes to chaos and bifurcation phenomena are demonstrated with examples. A brief discussion of equilibrium points and their stability is given. For the convenience of the reader, a short computer program written in QuickBASIC is included to give the reader a possibility of quick hands-on experience with the generation of chaotic phenomena without using sophisticated numerical simulators. All the necessary parameter values and initial conditions are provided in a tabular form. Eigenvalue diagrams showing regions with particular eigenvalue patterns are given.
The dynamical properties of two classical paradigms for chaotic behavior are reviewed—the Lorenz and Chua’s Equations—on a comparative basis. In terms of the mathematical structure, the Lorenz Equation is more complicated than Chua’s Equation because it requires two nonlinear functions of two variables, whereas Chua’s Equation requires only one nonlinear function of one variable. It is shown that most standard routes to cbaos and dynamical phenomena previously observed from the Lorenz Equation can be produced in Chua’s system with a cubic nonlinearity. In addition, we show other phenomena from Chua’s system which are not observed in the Lorenz system so far. Some differences in the topological geometric models are also reviewed. We present some theoretical results regarding Chua’s system which are absent for the Lorenz system. For example, it is known that Chua’s system is topologically conjugate to the class of systems with a scalar nonlinearity (except for a measure zero set) and is therefore canonical in this sense. We conclude with some reasons why Chua’s system can be considered superior or more suitable than the Lorenz system for various applications and studies.
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