1. 洛伦兹方程为：

   $$ \begin{cases} \frac{dx}{dt} = \sigma(y-x) \\ \frac{dy}{dt} = x(\rho-z)-y \\ \frac{dz}{dt} = xy-\beta z \end{cases} $$

   其中，$\sigma, \rho, \beta$ 为常数。

雅可比矩阵为：

$$ J = \begin{bmatrix} \frac{\partial f_1}{\partial x} & \frac{\partial f_1}{\partial y} & \frac{\partial f_1}{\partial z} \\ \frac{\partial f_2}{\partial x} & \frac{\partial f_2}{\partial y} & \frac{\partial f_2}{\partial z} \\ \frac{\partial f_3}{\partial x} & \frac{\partial f_3}{\partial y} & \frac{\partial f_3}{\partial z} \end{bmatrix} $$

其中，$f_1 = \sigma(y-x), f_2 = x(\rho-z)-y, f_3 = xy-\beta z$。

计算过程如下：

$$ J = \begin{bmatrix} -\sigma & \sigma & 0 \\ \rho-z & -1 & -x \\ y & x & -\beta \end{bmatrix} $$

2. 使用Matlab求解洛伦兹方程的代码如下：

   sigma = 10;
   rho = 28;
   beta = 8/3;
   dt = 0.01;
   t = 0:dt:50;
   x = zeros(size(t));
   y = zeros(size(t));
   z = zeros(size(t));
   x(1) = 1;
   y(1) = 1;
   z(1) = 1;
   for i = 2:length(t)
    x(i) = x(i-1) + sigma*(y(i-1)-x(i-1))*dt;
    y(i) = y(i-1) + (x(i-1)*(rho-z(i-1))-y(i-1))*dt;
    z(i) = z(i-1) + (x(i-1)*y(i-1)-beta*z(i-1))*dt;
   end
   plot(x,y)
   xlabel('x')
   ylabel('y')
   title('Lorenz Attractor')

   选取相对瑞利数为28，初始状态为$(1,1,1)$，得到的轨迹如下图所示：

3. 使用Matlab求解洛伦兹方程的代码如下：

   sigma = 10;
   rho = 28;
   beta = 8/3;
   dt = 0.01;
   t = 0:dt:50;
   x1 = zeros(size(t));
   y1 = zeros(size(t));
   z1 = zeros(size(t));
   x2 = zeros(size(t));
   y2 = zeros(size(t));
   z2 = zeros(size(t));
   x1(1) = 1;
   y1(1) = 1;
   z1(1) = 1;
   x2(1) = 1.01;
   y2(1) = 1.01;
   z2(1) = 1.01;
   for i = 2:length(t)
    x1(i) = x1(i-1) + sigma*(y1(i-1)-x1(i-1))*dt;
    y1(i) = y1(i-1) + (x1(i-1)*(rho-z1(i-1))-y1(i-1))*dt;
    z1(i) = z1(i-1) + (x1(i-1)*y1(i-1)-beta*z1(i-1))*dt;
    x2(i) = x2(i-1) + sigma*(y2(i-1)-x2(i-1))*dt;
    y2(i) = y2(i-1) + (x2(i-1)*(rho-z2(i-1))-y2(i-1))*dt;
    z2(i) = z2(i-1) + (x2(i-1)*y2(i-1)-beta*z2(i-1))*dt;
   end
   plot(t,x1,'b',t,x2,'r')
   xlabel('t')
   ylabel('x')
   title('Lorenz Attractor')
   legend('Initial state: (1,1,1)','Initial state: (1.01,1.01,1.01)')

   选取相对瑞利数为28，初始状态分别为$(1,1,1)$和$(1.01,1.01,1.01)$，得到的轨迹如下图所示：