Probabilistic control of the classical and quantum kicked rotor

The kicked rotor is a paradigmatic model for the study of chaos and integrability transitions in nonlinear dynamical systems. We introduce a probabilistic control scheme that couples the chaotic kicked rotor dynamics with a stochastic control map, enabling the stabilization of an otherwise unstable fixed point. By varying the control probability p, the system undergoes a control-driven phase transition: for p < p_cr, chaotic trajectories dominate, whereas for p > p_cr, chaos is fully suppressed and all trajectories converge to the fixed point. We characterize this transition using order parameter. From the finite-size scaling analysis, and Lyapunov exponent calculations, we extract critical probabilities for different kicked strength K and critical exponents. Extending to quantum dynamics, we implement control via measurements and feedback, and by coupling these with the unitary evolution of the quantum kicked rotor, we investigate how the control transition interplays with quantum localization.