We present a mapping of potentially chaotic time-dependent quantum kicked systems to an equivalent effective time-independent scenario, whereby the system is rendered integrable. The time-evolution is factorized into an initial kick, followed by an evolution dictated by a time-independent Hamiltonian and a final kick. This method is applied to the kicked top model. The effective time-independent Hamiltonian thus obtained, does not suffer from spurious divergences encountered if the traditional Baker-Cambell-Hausdorff treatment is used. The quasienergy spectrum of the Floquet operator is found to be in excellent agreement with the energy levels of the effective Hamiltonian for a wide range of system parameters. The density of states for the effective system exhibits sharp peak-like features, pointing towards quantum criticality. The dynamics in the classical limit of the integrable effective Hamiltonian shows remarkable agreement with the non-integrable map corresponding to the actual time-dependent system in the non-chaotic regime. This suggests that the effective Hamiltonian serves as a substitute for the actual system in the non-chaotic regime at both the quantum and classical level.
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