Lorenz Equation and Chua’s Equation

Abstract

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.

Structure of saddle-node and cusp bifurcations of periodic orbits near a non-transversal T-point

“…The existence of closed curves of global bifurcations (homoclinic and heteroclinic connections) in Chua's equation [28] has been reported numerically [4]. The mechanism of formation/destruction on such curves, when a third parameter is moved, is also qualitatively described and is related to a failure of transversality of a curve of T-points in a three-dimensional parameter space.…”

“…If we substitute the value of z given by equation (17) into (30), after some tedious algebra, it is possible to obtain an equivalent system, up to first order, to (27)- (28). This system is written in the following theorem:…”

Synchronization and Peak-to-Peak Dynamics in Networks of Low-Dimensional Chaotic Oscillators

“…The Chua oscillator The Chua circuit is described by the following state equations ( [Pivka et al, 1996]):ẋ…”

“…More precisely, in the next section we study pairs of coupled chaotic oscillators, but for three different types of oscillators, namely Lorenz system ( [Lorenz, 1963]), Rössler oscillator ( [Rössler, 1976]), and Chua circuit ( [Pivka et al, 1996]). Of course the analysis is performed by varying local coherence and coupling strength and by fixing parameter values which guarantee that these oscillators produce low-dimensional chaos when they are uncoupled.…”

A systematic procedure for synthesizing two‐terminal devices with polynomial non‐linearity

“…The study of circuits with polynomial non-linearity, especially of electronic chaos oscillators such as Chua's circuit, has been a topic of increasing interest for some time [1][2][3][4]. It was reported that not all features of a real circuit with a smooth non-linearity were correctly captured by the circuit with a piecewise-linear approximation.…”