Chaos and Non-Linear Dynamics Problem:

Consider the following differential equations:

x' = y z ...(1)

y' = x + ay ...(2)

z' = 2 + z(x 4) ...(3)

where x, y, z R and a 0 is a parameter.

(a) Find the equilibria of the system and give conditions for when they exist.

(b) By examining the characteristic equation, show that no = 0 bifurcations occur.

(c) Find the a-value where a Hopf bifurcation occurs.