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We analyze photometric data of the active RS CVn--type star HR 1099 for theyears 1975--2006 with an inversion technique and reveal the nature of twoactivity cycles of 15--16 yr and 5.3$\pm$0.1 yr duration. The 16 yr cycle isrelated to variations of the total spot area and is coupled with thedifferential rotation, while the 5.3 yr cycle is caused by the symmetricredistribution of the spotted area between the opposite stellar hemispheres(flip-flop cycle). We recover long-lived active regions comprising two activelongitudes that migrate in the orbital reference frame with a variable ratebecause of the differential rotation along with changes in the mean spotlatitudes. The migration pattern is periodic with the 16 yr cycle. Combiningthe longitudinal migration of the active regions with a previously measureddifferential rotation law, we recover the first stellar butterfly diagramwithout an assumption about spot shapes. We find that mean latitudes of activeregions at opposite longitudes change antisymmetrically in the course of the 16yr cycle: while one active region migrates to the pole, the other approachesthe equator. This suggests a precession of the global magnetic field withrespect to the stellar rotational axis.Comment: ApJL, accepte

Butterfly Diagram and Activity Cycles in HR 1099