  Fast scrambling of quantum correlations, reflected by the exponential growth
of Out-of-Time-Order Correlators (OTOCs) on short pre-Ehrenfest time scales, is
commonly considered as a major quantum signature of unstable dynamics in
quantum systems with a classical limit. In two recent works [Phys. Rev. Lett.
123, 160401 (2019)] and [Phys. Rev. Lett. 124, 140602 (2020)], a significant
difference in the scrambling rate of integrable (many-body) systems was
observed, depending on the initial state being semiclassically localized around
unstable fixed points or fully delocalized (infinite temperature).
Specifically, the quantum Lyapunov exponent $\lambda_{\rm q}$ quantifying the
OTOC growth is given, respectively, by $\lambda_{\rm q}=2\lambda_{\rm s}$ or
$\lambda_{\rm q}=\lambda_{\rm s}$ in terms of the stability exponent
$\lambda_{\rm s}$ of the hyperbolic fixed point. Here we show that a wave
packet, initially localized around this fixed point, features a distinct
dynamical transition between these two regions. We present an analytical
semiclassical approach providing a physical picture of this phenomenon and
support our findings by extensive numerical simulations in the whole parameter
range of locally unstable dynamics of a Bose-Hubbard dimer. Our results suggest
that the existence of this crossover is a hallmark of unstable separatrix
dynamics in integrable systems, thus opening the possibility to distinguish the
latter, on the basis of this particular observable, from genuine chaotic
dynamics generally featuring uniform exponential growth of the OTOC.
