Consider the two dimensional system............................................

Problem 3: (28 points) Consider the twodimensional system: 56 = yy2 3'; = sin($) a) Find an energy function ECE, y) of the form E(:1:, y) = a; cos(:1:)+ f (y), for some constant a and some function f. Verify that E(:c, y) is constant along trajectories. b) Show that the xed points ((271 + 1)7r, 0) and (27m, 1) are nonlinear centers. C) Show that the xed points (27m, 0) and ((271 + 1)7r, 1) are saddles. d) Sketch the phase portrait for this system