We have observed a new cataclysmic variable (CV) SDSS J080434.20+510349.2 and study the origin of a long-term variability found in its light curve. Multi-longitude time-resolved photometric observations were carried out to analyze the uncommon behavior also found recently in two newly discovered CVs. This study of SDSS J080434.20+510349.2 mainly concerns the understanding of the nature of the observed double-humped light curve and its relation to a cyclic brightening occurring during quiescence. The observations were obtained early in 2007, when the object was at about V~17.1, 0.4 mag brighter than the pre-outburst magnitude. The light curve shows a sinusoidal variability with an amplitude of about 0.07 mag and a periodicity of 42.48 min, which is half of the orbital period of the system. In addition, we have observed two mini-outbursts of the system up to 0.6 mag, with a duration of about 4 days each. The mini-outburst had a symmetric profile and repeated in about 32 days. Subsequent monitoring of the system shows a cyclical behaviour of such mini-outbursts with a similar recurrence period. The origin of the double-humped light curve and the periodic brightening is discussed in the light of the evolutionary state of SDSS J080434.20+510349.2.
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