1) Explain 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 characterise super and sub-critical pitchfork and hopf.(Give an example of each)
i) transcritical.
ii) pitchfork(super and sub-critical)
iii) Hopf(super and sub-critical)
iv) saddle-node
Use matlab or maple software.
How to starting the project main ideas: looks like story.
1) what it is ?
2) motivation/ applications.
3) Theory (Give related theorem and proof (some detail))./mechanics(how to find it and compute it algorithmic).
4) Given some example(one will do but given) (population biology research: used centre manifold theory to prove backward bifurcation).
5) Limitations/applicability Non- autonomous→poincare’ section.