4. (30 points) Consider the coupled oscillators 0 = cos(0 - 0) - 1 9 = 1 ( 1 + cos ( 0 - 0 ) ) where u can be positive, negative, or zero. (a) Let $ = 0 - 0 be the phase difference. Derive the 1-D system for p. (b) Draw the qualitatively different phase portraits that arise as / varies. You may use any types of plots to help you determine fixed points and stability, but present your final answers as phase portraits on the circle. (c) For what values of u is there phase drift? (d) For what values of / is there entrainment? (e) Find the period of the phase drift