Synchronization of spatiotemporal chaos in dissipative helicoidal spin–orbit coupled Bose–Einstein condensates

We investigate dissipative pattern formation in helicoidal spin–orbit (SO) coupled Bose–Einstein condensates (BECs). By numerically solving the full two-component complex Ginzburg–Landau equations, we uncover a rich variety of spatiotemporal dynamics, including tree-like branching patterns, localized hole-like structures, and shock-like features. These patterns exhibit strong qualitative similarities to turbulent regimes of the one-dimensional complex Ginzburg–Landau equation. We show that the system evolves from phase turbulence toward defect-dominated chaos through an intermediate intermittency regime, where localized hole-shock structures appear. Furthermore, the combined effects of SO coupling and the helicoidal gauge potential provide an effective coupling mechanism between the two components, leading to strongly correlated dynamics and, in certain regimes, near-synchronous spatiotemporal behavior even in the presence of chaos. These results highlight how complex and coordinated structures can emerge from intrinsically nonlinear and dissipative dynamics in non-equilibrium quantum fluids.