Question

For the Hamiltonian syste m we did in class: 2. 3 Ic (1) Show that it's a Hamiltonian system with a Hamiltonian function (2) Show that for each c > 0, {(z,y) є R2 . H(z,y) c} is a bounded invariant set of the dynamical system (in fact, it's also closed) (3) Find all the equilibria of this system. Show that H-() is made up of one equilibium point and two homoclinic orbits attached to it. (4) Sketch the invariant set {(x,y) E R2 : H(z,y)-c} on the xy-plane. You may divide the cases into c 0, 0 < cく름' c = 1, and c > 1. (5) Using all the information above, Poincaré-Bendixon theorem and its corollaires to show that all other trajectories (than those you found in part (3)) are cycles 4

Answer 1

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