Problem (LORENZ)

Consider the Lorenz system of differential equations: $$\begin{align*} \frac{dx}{dt} &= \sigma (y - x) \\ \frac{dy}{dt} &= x (\rho - z) - y \\ \frac{dz}{dt} &= x y - \beta z \end{align*}$$ where \(\sigma = 10\), \(\rho = 28\), and \(\beta = 8 / 3\) are positive parameters.

1. Find the equilibrium points of the Lorenz system.
2. Linearize the system around each equilibrium point.
3. Analyze the stability of each equilibrium point using the linearized system.
4. Discuss the implications of your findings for the behavior of the Lorenz system.
5. Numerically simulate the Lorenz system for various initial conditions and visualize the trajectories in 3D space.