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Non-KAM (Kolmogorov-Arnold-Moser) systems, when perturbed by weaktime-dependent fields, offer a fast route to classical chaos through an abruptbreaking of invariant phase space tori. In this work, we employout-of-time-order correlators (OTOCs) to study the dynamical sensitivity of aperturbed non-KAM system in the quantum limit as the parameter thatcharacterizes the $\textit{resonance}$ condition is slowly varied. For thispurpose, we consider a quantized kicked harmonic oscillator (KHO) model, whichdisplays stochastic webs resembling Arnold's diffusion that facilitatelarge-scale diffusion in the phase space. Although the Lyapunov exponent of theKHO at resonances remains close to zero in the weak perturbative regime, makingthe system weakly chaotic in the conventional sense, the classical phase spaceundergoes significant structural changes. Motivated by this, we study the OTOCswhen the system is in resonance and contrast the results with the non-resonantcase. At resonances, we observe that the long-time dynamics of the OTOCs aresensitive to these structural changes, where they grow quadratically as opposedto linear or stagnant growth at non-resonances. On the other hand, our findingssuggest that the short-time dynamics remain relatively more stable and show theexponential growth found in the literature for unstable fixed points. Thenumerical results are backed by analytical expressions derived for a fewspecial cases. We will then extend our findings concerning the non-resonantcases to a broad class of near-integrable KAM systems.

Probing Dynamical Sensitivity of a Non-KAM System Through Out-of-Time-Order Correlators