HOMEWORK10

author : 2013301020086 Yuqi Wang
homework

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1.Introduction

E.N.Lorenz (1963) was studying the basic equations of fluid mechanics, ehich are known as the Navier-Stokes equitions; they can be thought of as Newton's laws written in a form appropriate for a fluid. These are a complicated set of differential equations that describe the celocity, tempreture, density, etc., as functions of position and time, and that is precisely what Lorenz did. indeed, he grossly oversimplified the problem as he reduced it to only three equitions:

$$\frac{dx}{dt}=\sigma(y-x)$$
$$\frac{dy}{dt}=-xz+rx-y$$
$$\frac{dz}{dt}=xy-bz$$
which are now known as Lorenz equitions. The Lorenz variables $x,y$ and $z$ are derived from the tempreture, density, and velocity variables in original Naviar-Stokes equitions, and the parameters $\sigma,r$ and $b$ are measures of the tempreture difference across the fluid and other fluid parameters. Here I choose $\sigma=10,b=\frac{8}{3}$, and the system will behave in different ways with changes over $r$.

2.Method and Result

This is the complete code
Euler-Cromer method is applied:

$$v_{xi+1}=\sigma(y_i-x_i)$$
$$v_{yi+1}=-x_iz_i+rx_i-y_i$$
$$v_{zi+1}=x_iy_i-bz_i$$
$$x_{i+1}=x_i+v_{xi+1}dt$$
$$y_{i+1}=y_i+v_{yi+1}dt$$
$$z_{i+1}=z_i+v_{zi+1}dt$$

With the initial condition $x=1,y=0,z=0$, I get: