human
I am working on the file "Fall2018/07-Jupyter-Notebook-In-Depth/LorenzSystem.ipynb".
The first task is:
Computing the trajectories and plotting the result
We define a function that can integrate the differential equations numerically and then plot the solutions. This function has arguments that control the parameters of the differential equation (\(\sigma\), \(\beta\), \(\rho\)), the numerical integration (N, max_time) and the visualization (angle).
Can you write Python code for it?

Write me a the Python code you requested?
assistant
#Here is the Python code you requested.

def solve_lorenz(N=10, angle=0.0, max_time=4.0, sigma=10.0, beta=8./3, rho=28.0):

    fig = plt.figure()
    ax = fig.add_axes([0, 0, 1, 1], projection='3d')
    ax.axis('off')

    # prepare the axes limits
    ax.set_xlim((-25, 25))
    ax.set_ylim((-35, 35))
    ax.set_zlim((5, 55))
    
    def lorenz_deriv(x_y_z, t0, sigma=sigma, beta=beta, rho=rho):
        """Compute the time-derivative of a Lorenz system."""
        x, y, z = x_y_z
        return [sigma * (y - x), x * (rho - z) - y, x * y - beta * z]

    # Choose random starting points, uniformly distributed from -15 to 15
    np.random.seed(1)
    x0 = -15 + 30 * np.random.random((N, 3))

    # Solve for the trajectories
    t = np.linspace(0, max_time, int(250*max_time))
    x_t = np.asarray([integrate.odeint(lorenz_deriv, x0i, t)
                      for x0i in x0])
    
    # choose a different color for each trajectory
    colors = plt.cm.viridis(np.linspace(0, 1, N))

    for i in range(N):
        x, y, z = x_t[i,:,:].T
        lines = ax.plot(x, y, z, '-', c=colors[i])
        plt.setp(lines, linewidth=2)

    ax.view_init(30, angle)
    plt.show()

    return t, x_t