Close to the origin, where x, y, and z are very small, the quadratic terms -xz and +xy wil...

Close to the origin, where x, y, and z are very small, the quadratic terms -xz and +xy will be very, very small. So, near (x, y, z) = (0, 0, 0), we can approximate the Lorenz system with the system

(This is called linearization at the origin.) Notice that z does not appear in the equations for dx/dt and dy /dt, and the equation for dz/dt does not contain x or y. That is, the system decouples into a two dimensional system and a one-dimensional equation.

(a) Using HPGSystemSolver, sketch the direction field and the phase plane for the planar system

(b) Sketch the phase line for the equation

(c) Sketch solutions in the three-dimensional phase space for the system above. (This picture gives the behavior of the Lorenz system near (0, 0, 0).)