Context. The relatively rapid spatial and temporal variability of the X-ray radiation from some molecular clouds near the Galactic center shows that this emission component is due to the reflection of X-rays generated by a source that was luminous in the past, most likely the central supermassive black hole, Sagittarius A*. Aims: Studying the evolution of the molecular cloud reflection features is therefore a key element in reconstructing Sgr A*'s past activity. The aim of the present work is to study this emission on small angular scales in order to characterize the source outburst on short time scales. Methods: We use Chandra high-resolution data collected from 1999 to 2011 to study the most rapid variations detected so far, those of clouds between 5' and 20' from Sgr A⋆ toward positive longitudes. Our systematic spectral-imaging analysis of the reflection emission, notably of the Fe Kα line at 6.4 keV and its associated 4-8 keV continuum, allows us to characterize the variations down to a 15'' angular scale and a 1-year time scale. Results: We reveal for the first time abrupt variations of only a few years and, in particular, a short peaked emission, with a factor of 10 increase followed by a comparable decrease, which propagates along the dense filaments of one cloud. This 2-year peaked feature contrasts with the slower 10-year linear variations we reveal in all the other molecular structures of the region. Based on column density constraints, we argue that these two different behaviors are unlikely to be due to the same illuminating event. Conclusions: The variations likely stem from a highly variable active phase of Sgr A⋆ which occurred sometime within the past few hundred years, and is characterized by at least two luminous outbursts with typical time scales of a few years and during which the Sgr A⋆ luminosity went up to at least 1039 erg s-1
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