SUMMARY

Having completed this unit you should

• Be able to linearise a nonlinear system
• Be able to classify the equilibrium points of the linear system
• Know that a nonlinear system mimics its linearisation close to an equilibrium point unless the linearisation is nonsimple or a centre.
• Know that the existence of a nonlinear centre can be verified if the system is conservative and the equilibrium point is a local maximum or minimum of the conserved quantity.
• Know that a first integral is a conserved quantity if it exists for all (x ,y).

• Know that the existence of a nonlinear centre can be verified if the system is reversible.
• Know that global stability can be determined in a conservative system and the solutions of the system are the same as the level curves of the conserved quantity E(x,y).
• Know that global stability can be determined if a Lyapunov function can be found for the system and the solutions of the system cross the level curves L(x,y) =C in the direction of decreasing C