A computer proof that the Lorenz equations have “chaotic” solutions

Chaos in the Lorenz Equations: a Computer-assisted Proof

A new technique for obtaining rigorous results concerning the global dynamics of nonlinear systems is described. The technique combines abstract existence results based on the Conley index theory with computer-assisted computations. As an application of these methods it is proven that for an explicit parameter value the Lorenz equations exhibit chaotic dynamics. Introduction The purpose of this...

Exploring the Lorenz Equations through a Chaotic Waterwheel

were derived by Edward Lorenz in 1963 as an over-simplified model of convection rolls within the atmosphere [1]. These coupled differential equations appear quite harmless; only two terms appear that cause non-linearity. Lorenz found, however, that the behavior of the system changed drastically over a wide range of parameters. For certain parameter values, Lorenz discovered that the system exhi...

Chaos in the Lorenz Equations : A Computer Assisted Proof ; Part III : Classical Parameter

Chaos in the Lorenz equations: A computer assisted proof. Part II: Details

Details of a new technique for obtaining rigorous results concerning the global dynamics of nonlinear systems is described. The technique combines abstract existence results based on the Conley index theory with rigorous computer assisted computations. As an application of these methods it is proven that for some explicit parameter values the Lorenz equations exhibit chaotic dynamics.

Stabilization of the Lorenz Chaotic Equations by Fuzzy Controller

In this paper, a fuzzy controller is designed for stabilization of the Lorenz chaotic equations. A simple Mamdani inference method is used for this purpose. This method is very simple and applicable for complex chaotic systems and it can be implemented easily. The stability of close loop system is investigated by the Lyapunov stabilization criterion. A Lyapunov function is introduced and the gl...