Bundles of actin filaments and molecular motors of the myosin family are a common subcellular organizational motif. Typically, such bundles are under contractile stress resulting from interactions between the filaments and the motors. This holds in particular for contractile rings that appear in the late stages of cell division in animal cells and that cleave the mother into two daughter cells. It was recently shown that myosin organizes into regularly spaced clusters along rings in mammalian cells, whereas myosin clusters in fission yeast travel along the perimeter of actomyosin rings [Wollrab et al., Nat. Commun. 7, 11860 (2016)]. A mechanism based on the association of the structurally polar actin filaments into bipolar structures was shown to provide a common explanation for both observations. Here, we analyze the dynamics of this mechanism in detail. We find a rich phase diagram depending on the actomyosin interaction strength and the stability of the bipolar structures. The system can notably organize into traveling waves. Furthermore, we identify the nature of the bifurcations connecting the various patterns as parameters are changed. Finally, we report experimental patterns observed in cytokinetic rings in fission yeast and link them to solutions of our dynamic equations. Our analysis highlights the possible role played by local polarity sorting of actin filaments for the dynamics and functionality of actomyosin networks.