Consider techno music, or the hum of alternating current. Whether due to nature, or imposed artificially by humans, cyclic behavior can be found everywhere. As these behaviors are often interesting to study and analyze, cyclic data becomes as popular as the behavior it models. Often the data is compiled into a time series in which the data is cyclic with respect to time. Ever more frequently, interesting data is multivariate, that is, varying in multiple dimensions.
Cyclic data is no exception.
Consider each of the examples in the paragraph above. Many variables may be associated with each cycle. We may be interested not only in an ocean wave's position with respect to time, but also its amplitude, velocity, temperature, and salinity, some of which may also be cyclic. Multivariate data may have variables cyclic with respect to the same dimension, or may have variables cyclic with respect to different dimensions, or may not be cyclic at all. If cyclic with respect to the same dimension, variables still may be out of phase or have different cycle lengths, or periods.