Equivariant normal form for Turing-Hopf bifurcation and superposition of rotating waves and spatial supersquare patterns on 2D square domains

We explore pattern formation (rotating waves, spatial supersquare/square patterns, periodic oscillators) and their superposition/coexistence on a 2D square domain via equivariant Turing-equivariant Hopf (ET-EH) bifurcation. We first establish the third-order equivariant normal form of ET-EH bifurcation for nonlocal partial functional differential equations (PFDEs), presenting two simplified forms (selected from 43 systematically classified cases; full details of the 43 cases are given in Appendix E ) and concise formulae for their coefficients. Notably, the proposed explicit algorithm—using original system parameters to compute the third-order normal form—applies to both ET-EH and TH bifurcations. Using this normal form, we analyze a nonlocal Holling-Tanner model near ET-EH singularity, theoretically predicting and numerically demonstrating diverse patterns and coexistence: bistable clockwise/anticlockwise rotating waves (modes (1,0),(0,1)), quad-stable spatial supersquare patterns (modes (1,2),(2,1)), and their superposition, as well as periodic oscillators (mode (0,0)), bistable square patterns (mode (1,1)), and their coexistence.