Unfolding spatiotemporal dynamics through symmetry reduction based on orbit topology

Understanding key patterns in a spatially extended system is an essential task of modern physics of complex systems. Just like in low-dimensional nonlinear systems, here we show that orbit topology plays a critical role even for the investigation of spatiotemporal dynamics. First, we design a new scheme to reduce possible continuous symmetries that are prevailing in these systems based on topological consideration. The scheme is successfully demonstrated in the well-known pattern formation systems. Interesting bifurcation routes to chaos are conveniently revealed after symmetry reduction. In particular, we find that near the onset of turbulent dynamics, with an increase of instability, local phase chaos with the same spatial topological index may merge into more complex ones, while those with different indices induce defect chaos necessarily through connections docked with defects. The topological argument is so strong that the scenario presented here should be omnipresent in diverse systems.