Separatrices Splitting for the Rapidly Forced Pendulum

The equation of the rapidly forced pendulum % MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 % qaceWG4bGbamaacqGHRaWkciGGZbGaaiyAaiaac6gacaGGOaGaamiE % aiaacMcacqGH9aqpcqaH0oazcqaH1oqzpaWaaWbaaSqabeaapeGaam % iCaaaakiGacohacaGGPbGaaiOBaiaacIcacaWG0bGaai4laiabew7a % LjaacMcacaGGSaaaaa!4AFC! $$\ddot x + \sin (x) = \delta {\varepsilon ^p}\sin (t/\varepsilon ),$$ p > 5, is considered. It is proven that Poincaré-Arnold-Melnikov method gives the right asymptotic formula for the separatrices splitting angle for the case of a small ε.