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4. Existence of periodic solutions (a) Briefly explain the conditions under which the Poincar-Bendixson theorem for two-dimensional dynamical systems guarantees the existence of a periodic orbit. (b) Use the Poincar-Bendixson theorem to discuss the existence of a limitcycle for the system d r (1 12) + ur? cos (20) dt exp (r) + sin (20) dt For which values of u do you expect that a limit cycle exists? Do you expect it to be stable? (c) Show that the system d = -2-y+cY d = y + 2x3 - 2.4 has no periodic solutions. Hint: show that V 20M + ay" is a Lyapunov function for suitable m, n, a. 2 4. Existence of periodic solutions (a) Briefly explain the conditions under which the Poincar-Bendixson theorem for two-dimensional dynamical systems guarantees the existence of a periodic orbit. (b) Use the Poincar-Bendixson theorem to discuss the existence of a limitcycle for the system d r (1 12) + ur? cos (20) dt exp (r) + sin (20) dt For which values of u do you expect that a limit cycle exists? Do you expect it to be stable? (c) Show that the system d = -2-y+cY d = y + 2x3 - 2.4 has no periodic solutions. Hint: show that V 20M + ay" is a Lyapunov function for suitable m, n, a. 2