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Dynamical systems often contain oscillatory forces or depend on periodicpotentials. Time or space periodicity is reflected in the properties of thesesystems through a dependence on the parameters of their periodic terms. In thispaper we provide a general theoretical framework for dealing with these kindsof systems, regardless of whether they are classical or quantum, stochastic ordeterministic, dissipative or nondissipative, linear or nonlinear, etc. Inparticular, we are able to show that simple symmetry considerations determine,to a large extent, how their properties depend functionally on some of theparameters of the periodic terms. For the sake of illustration, we apply thisformalism to find the functional dependence of the expectation value of themomentum of a Bose-Einstein condensate, described by the Gross-Pitaewskiiequation, when it is exposed to a sawtooth potential whose amplitude isperiodically modulated in time. We show that, by using this formalism, a smallset of measurements is enough to obtain the functional form for a wide range ofparameters. This can be very helpful when characterizing experimentally theresponse of systems for which performing measurements is costly or difficult.

General approach for dealing with dynamical systems with spatiotemporal periodicities