????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????? ? ???  ?   ?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.

