No, the storms on Jupiter aren't oscillating forth and back.
The animated gif is showing the southern cluster of Jupiter's circumpolar cyclones (CPCs). It is a 6000-fold timelapsed animation running forth and back in time, in order to look smoother for the human eye.
Four of the CPCs are well visible, two are located deep in the twilight zone, and difficult to resolve. Note the varying symmetrical properties of the CPC dynamics. Some rotate pretty rotational symmetric, others are struggling with embedded storm systems, some rotate like a solid body around their core, others rotate the higher frequency the closer to their center.
The gap between two of the CPCs has opened to more than 120 degrees, and it is now populated by a poorly resolved anticyclone close to the central CPC, and a small roundish filamentary folded region (FFR), candidate for a young CPC, which might eventually complete a CPC hexagon around the central southern CPC.
The southern cluster of CPCs is surrounded by many cyclonic and anticyclonic storms.
Note the shading of the bluish haze near the terminator moving with changing illumination.
Each of the 20 frames of the animation is a weighted mean of two images, each of which is stacked from three heavily processed equidistant azimuthal south polar maps. The image noise level varying with color was considered. The animation is based on a set of ten raw JunoCam images taken during Juno's Perijove-18 flyby.
The scale of the animated map is 60 pixels per planetocentric degree latitude.