1. Discuss lyapunov function theory and how it can be used to prove global assmptotic stability of solutions.(Give an example form natural and engineering sciences.)

--- Draw level sets of Lyapunov function.

2. Discuss Perron-Frobenius theorem and how it can be used to prove global stability of solutions(Give an example) .

3. Discuss the stability of solutions of non-autonomous system dx/dt=f(x,t).

4. Discuss bifurcation theory, and show how to (theoretically) characterise varies, bifurcation types apriori (saddle node, pitchfork, hopf, transcritical). Show to charactrise super and sub-critical pitchfork and hopf.(Give an example of each)

1) transcritical.

2)pitchfork(super and sub-critical)

3)Hopf(super and sub-critical)

4)saddle-node