1 For each of the following systems 

a) show 

 EMBED Equation  


		is a Lyapunov function for the system.

b) show  EMBED Equation  
is an equilibrium point of the system.

c) show  EMBED Equation  
is a stationary point for  EMBED Equation  


d) determine the type of stationary point  by looking at the eigenvalues of the Hessian matrix

e) determine the stability of the equilibrium point.



  	A)    EMBED Equation  
	 EMBED Equation  


	  B)    EMBED Equation  
	 EMBED Equation  


  C)    EMBED Equation  
	 EMBED Equation  
	

     

2 Repeat question (1) for

 EMBED Equation  


and the following systems

	a)    EMBED Equation  
	 EMBED Equation  


b)    EMBED Equation  
	 EMBED Equation  




3 Show that the system

 EMBED Equation  


has no closed orbits by construsting a Lyapunov function 

 EMBED Equation  


with suitable a and b.