Nonlinear frequency response of the betatron oscillation at the photon factory storage ring

Octupole magnet takes an important part in the cure of the transverse instability at the Photon Factory storage ring. Under the existence of the octupole field, remarkable asymmetric frequency-response curves were observed in the excitation of the vertical beratron oscillation. The form of the response curve depended on the octupole current and the direction of the frequency scan. The betatron resonance continued much above the betatron tune and the amplitude of the oscillation suddenly fall down at the end. The frequency response extended to about 20 kHz higher than the tune frequency when the maximum octupole field was applied. Such asymmetric frequency responses appeared only for the positive frequency scan. We have found that the observed frequency response for the betatron oscillation has many similarities with the frequency response of the nonlinear forced oscillation described by the Duffing equation. (author)