Chaos and strange attractors of the lorenz equations

Abstract

This research introduces and analyzes the famous Lorenz equations which are a classical example of a dynamical continuous system exhibiting chaotic behavior. This system is a three-dimensional system of first order autonomous differential equations and their dynamics are quite complicated. Some basic dynamical properties, such as stability, bifurcations, chaos and attractor are studied, either qualitatively or quantitatively. The visualization of the strange attractor and chaotic orbit are displayed using phase portrait and also the time series graph. A way to detect the chaotic behavior of an orbit is by using the Lyapunov exponents which indicate chaoticity if there is at least one positive Lyapunov exponent. The Lyapunov dimension called Kaplan-Yorke dimension of the chaotic attractor of this system is calculated to prove the strangeness by non-integer number. Several visualization methods are applied to this system to help better understand the long time behavior of the system. This is achieved by varying the parameters and initial conditions to see the kind of behavior induced by the Lorenz equations. The mathematical algebra softwares, Matlab and Maple, are utilized to facilitate the study. Also, the compound structure of the butterfly-shaped attractor named Lorenz attractor is also explored.