Smirnov, A.V. JACoW Geneva, Switzerland 978-3-95450-180-9 10.18429/JACoW-NAPAC2016-TUPOB41 English 585-587 TUPOB41 ion polarization dipole experiment linac Contribution to a conference proceedings 2017 2017-01 http://dx.doi.org/10.18429/JACoW-NAPAC2016-TUPOB41 https://jacow.org/napac2016/papers/tupob41.pdf Transverse instability of a multi-bunch beam in the presence of a longitudinal magnetostatic field and hybrid dipole modes is considered analytically within a single-section model. It incorporates resonant interaction with beam harmonics and eigenmodes, degenerated waves of different polarizations, and the Lorentz RF force contribution. The analysis is performed in a very compact form using a bi-complex i,j-space including four-component collective frequency of the instability. Rotating polarization of the collective field is determined by ImiImj part of the bi-complex collective frequency in agreement with available data. The other three components represent detuning of the collective frequency ReiRej, the left-hand, and right-hand increments ImiRej±ReiImj of the gyro-magnetic BBU effect. The scalar hyper-complex toolbox can be applied to designing of non-ferrite non-reciprocal devices, spin transport, and for characterization of complex transverse dynamics in gyro-devices such as Gyro-TWTs.