the Lorenz nodel

homework of Computational Physics

张强 2013301020039

Code

Lorenz.py

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.

Key word

Lorenz system,chaos

Background

The exerxise 3.29ask us to explore the intermittency route to chaos for $r \geq 163$ in more details.

Body

1Overview

In 1963,Edward Lorenz developed a simplified mathematical model for atmospheric convection. The model is a system of three ordinary differential equations now known as the Lorenz equations:

$$\begin{align} \frac{\mathrm{d}x}{\mathrm{d}t} &= \sigma (y - x), \\ \frac{\mathrm{d}y}{\mathrm{d}t} &= x (\rho - z) - y, \\ \frac{\mathrm{d}z}{\mathrm{d}t} &= x y - \beta z. \end{align}$$

From a technical standpoint, the Lorenz system is nonlinear three-dimensional and deterministic system. The Lorenz equations have been the subject of hundreds of research articles, and at least one book-length study.

2.Euler method

Results(figure)

1.

Variation of the Lorenz variable $z$ as a function of time.The calculation was performed using the Euler method.

2.

The trajectory of the Lorenz model projected onto the $x-z$ plane,with defferent $r$.

Connclusion

Acknowledgement

Thanks to Chwn yangyao for his template.