@rfhongyi 2016-11-19T07:20:06.000000Z 字数 1869 阅读 625

Abstract

The Lorenz system is a system of ordinary differential equations (the Lorenz equations, note it is not Lorentz) first studied by Edward Lorenz. It is notable for having chaotic solutions for certain parameter values and initial conditions. In particular, the Lorenz attractor is a set of chaotic solutions of the Lorenz system which, when plotted, resemble a butterfly or figure eight. In this paper, I will use Lorenz modle to grasp a basic understanding of fluid mechanics, especially Rayleigh-Benard problem. Besides, phase-space plot, Poincare section are investigated.

Background

Lorentz was studying the basic equations of fluid mechanics, which are known as the Navier-Stokes equations and which can be regarded as Newton’s law written in a form appropriate to a fluid. The specific situation he considered was the Rayleigh-Benard problem, which concerns a fluide in a container whose top and bottom surfaces are held at different temperatures. Indeed, he grossly simplified the problem as he reached to the so-called Lorentz equations, or equivalently, the Lorentz model, which provided a major contribution to the modern field of physics.
The Lorentz equations (the Lorentz model):

$\frac{dx}{dt}=\sigma(y-x)$

$\frac{dy}{dt}=-xz+rx-y$

$\frac{dz}{dt}=xy-bz$
The Lorentz variables $x,y,z$ are derived from the temperature, density and velocity variables in the original Navier-Stokes equations, and the parameters $\sigma,r,b$ are measures of the temperature difference across the fluid and other fluid parameters.
Here, the Euler algorithm can actually be used to treat the Lorentz equation:

$x_{i+1}=x_i+\sigma(y_i-x_i)dt$

$y_{i+1}=y_i+(-x_iz_i+rx_i-y_i)dt$

$z_{i+1}=z_i+(x_iy_i-bz_i)dt$
In this report, the custom is followed and $\sigma=10,b=\frac{8}{3}$ is used. The parameter is a measure of the temperature difference between the top and the bottom of the fluid. In fact, plays the role analogous to the drive amplitude.