Systematic generation of the cascade of anomalous dynamical first- and higher-order modes in Floquet topological insulators

After extensive investigation on the Floquet second-order topological insulator (FSOTI) in two dimension (2D), here we propose driving schemes to systematically engineer hierarchy of first-order insulator, FSOTI, and third-order three (3D). Our protocols allow these phases showcase regular $0$, anomalous $\pi$, hybrid $0$-$\pi$-modes a unified phase diagram, obtained for both 2D 3D systems, while staring from lower order or non-topological phases. Both step drive mass kick exhibit analogous structure evolution operator around high symmetry points. These eventually enable us understand diagrams analytically higher modes numerically based finite size systems. The number $0$ $\pi$-modes can be tuned irrespective frequency scheme observe driven transitions protocol. We topologically characterize some (harboring either $\pi$ mode) by suitable invariant cases.