Superconductor/ferromagnet/superconductor Josephson junctions with anomalous phase shift $\varphi_0$ ($\varphi_0$-S/F/S JJs) realize a coupling between the superconducting phase and the spin degrees of freedom. Consequently, the anomalous phase shift is a hallmark effect of superconducting spintronics and opens great perspectives for applications of such structures for controlling magnetization. Besides controlling the magnetization of a single ferromagnet, the direct coupling between the phase and the magnetization provides a possibility to establish and control a collective behavior of magnetic moments of different weak links in coupled chains of $\varphi_0$-S/F/S JJs. Here we investigate voltage-driven dynamics of such chains and predict that the spin-phase coupling makes the conventional voltage-driven regime unstable. As a result the system undergoes a transition to other dynamical regimes, where clear signatures of the spin-phase coupling and the collective behavior are observed.
