We investigate topological phase transitions in a two-dimensional electron system with cubic Rashba spin-orbit coupling driven by circularly polarized light. Within the Floquet framework, we demonstrate that light-matter interaction induces nontrivial band topology characterized by a quantized anomalous Hall response, with Chern insulating phases of $\mathcal{C}=0,\pm 1,$ and $\pm 3$. These transitions are governed by gap closings at high-symmetry points in the Brillouin zone, controlled by intensity and energy of the incident light. Introducing a weak linear Rashba term displaces Dirac points in momentum space without modifying the topology, whereas a purely linear Rashba system remains topologically trivial $(\mathcal{C}=0)$. When both linear and cubic Rashba couplings are finite, the linear term confines nonzero-Chern phases to narrow parameter windows. In contrast, incorporating a linear Dresselhaus term into the cubic Rashba system can trigger topological transitions even at small coupling strengths. These results clarify the interplay between distinct spin-orbit couplings in Floquet-engineered Chern insulators and offer experimentally relevant pathways for achieving light-controlled topological phases.

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• Quantum anomalous Hall effect
• Spin-orbit coupling
• 2-dimensional systems
• Floquet systems
• Bloch-Floquet theorem