???????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????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.