(a) Find the eigenvalues of the following system and sketch a phase portrait in three-dimensional space:

dx dt = −2x − z, dy dt = −y, dz dt = x − 2z.

[12]

(b) Show that the origin of the following nonlinear system is not hyperbolic:

dx dt = −2y + yz, dy dt = x − xz − y3, dz dt = xy − z3.

Prove that the origin is asymptotically stable using the Lyapunov function V = x2 + 2y2 + z2. What does asymptotic stability imply for a trajectory γ (t) close to the origin?

[8]