????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????? ? ???  ?   ?b(o???? ????    ? ? ???? ? ???  ?   ?? ? ???   ?   ?b(? ? ????   ? ? ???? ? ???  ?   ?? ? ?????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????   ? ? ???? ? ???  ?   ?? ? ???  ?   ?b(o???? ????    ? ? ???? ? ???  ?   ?? ? ???   ?   ?b(? ? ????   ????????????????????????????????????????????b) ????????????????????????????????????????????????????????????????????????????????????? ?   ?b?? ? ????   ? ? ???? ? ???c)   ?   ?? ? ???  ?   ?b(o???? ????    ? ? ???? ? ???  ?   ?? ? ???   ?   ?b(???????????????????????????????????????????????d) ?????????????????????????????????????????????k that youre answers to part (c)??????? ? ??2 ?  ?   ?? ? ???   ?   ?b?? ? ????   ? ? ???? ? ???  ?   ?? ? ???  ?   ?b(o???? ????    ? ? ???? ? ?um points of the nonlinear system
 EMBED Equation  
	are  EMBED Equation  .
b) Using the Linearisation Theorem show that one equilibrium point is a nonlinear saddle but the other point needs further investigation.
c) By finding a First Integral find a conserved quantity for the system.
d) By finding the Hessian matrix show the unclassified equilibrium point is a minimum stationary point for the conserved quantity and is, therefore a nonlinear centre.
e) Plo t a phase portrait for the system.
Note: the existence of the nonlinear saddle could also have been proved using the Hessian matrix. Check this yourself.

3 Repeat question (3) for the system
 EMBED Equation  


1 Prove the following systems are reversible and use Maple to plot the phase portrait.
	    a)   EMBED Equation  	 EMBED Equation  
	    b)   EMBED Equation  	 EMBED Equation  
    c)   EMBED Equation  	 EMBED Equation  

2 Consider the system defined by 
 EMBED Equation  
	where f is an even and both f and g are differentiable. Show that
a) the system is invariant under time reversal symmetry  EMBED Equation  ;
b) the equilibrium points cannot be nodes or foci.
c) Illustrate the results by considering the systems 
 EMBED Equation  

1 For the system
 EMBED Equation  
a)  find the equilibrium point, linearise the system at the equilibrium point and
show that the equilibrium point of the linearisation is a centre.
b) Prove the system is reversible and hence classify the equilibrium point of the nonlinear system.

2 Repeat question (1)  for the system
 EMBED Equation  

4 Consider the system
 EMBED Equation  
a) Write the system as a pair of nonlinear differential equations.
b) Show the system has an infinity of equilibrium points at  EMBED Equation  .
c) By using the Linearisation theorem and proving the system reversible show that the equilibrium points are alternately nonlinear saddles and nonlinear centres.

1 Verify that 
 EMBED Equation  
is a Lyapunov function for the system 
 EMBED Equation  
2 Repeat question (1) for 
 EMBED Equation  
and the system
 EMBED Equation  

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.
