We have carried out multi-epoch, time-series differential I-band photometry of a large sample of objects in the south-east region of the young (~3 Myr), nearby (~350 pc) sigma Orionis open cluster. A field of ~1000 arcmin^2 was monitored during four nights over a period of two years. Using this dataset, we have studied the photometric variability of twenty-eight brown dwarf cluster candidates with masses ranging from the stellar-substellar boundary down to the planetary-mass domain. We have found that about 50% of the sample show photometric variability on timescales from less than one hour to several days and years. The amplitudes of the I-band light curves range from less than 0.01 up to ~0.4 magnitudes. A correlation between the near-infrared excess in the K_s band, strong Halpha emission and large-amplitude photometric variation is observed. We briefly discuss how these results may fit the different scenarios proposed to explain the variability of cool and ultracool dwarfs (i.e. magnetic spots, patchy obscuration by dust clouds, surrounding accretion discs and binarity). Additionally, we have determined tentative rotational periods in the range 3 to 40 h for three objects with masses around 60 M_Jup, and the rotational velocity of 14+/-4 km/s for one of them.
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