  We study universal chaotic dynamics of a large class of periodically driven
critical systems described by spatially inhomogeneous conformal field theories.
By employing an effective curved spacetime approach, we show that the onset of
quantum chaotic correlations, captured by the Lyapunov exponent of
out-of-time-order correlators (OTOCs), is set by the Hawking temperature of
emergent Floquet horizons. Furthermore, scrambling of quantum information is
shown to be strongly inhomogeneous, leading to transitions from chaotic to
non-chaotic regimes by tuning driving parameters. We finally use our framework
to propose a concrete protocol to simulate and measure OTOCs in quantum
simulators, by designing an efficient stroboscopic backward time evolution.
