We investigate dissipative dynamical systems under the influence of an external driving with two or more frequencies. Our main quantities of interest are long-time averages of expectation values which turn out to exhibit universal features. In particular, resonance peaks in the vicinity of commensurable frequencies possess a generic enveloping function whose width is inversely proportional to the averaging time. While the universal features can be derived analytically, the transition from the specific short-time behavior to the long-time limit is illustrated for the examples of a classical random walk and a dissipative two-level system both with biharmonic driving. In these models, the dependence of the time-averaged response on the relative phase between the two driving frequencies changes with increasing integration time. For short times, it exhibits the 2π periodicity of the dynamic equations, while in the long-time limit, the period becomes a fraction of this value.
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Olivera-Atencio, M.L., Casado-Pascual, J. & Kohler, S. Generic shape of multichromatic resonance peaks. Eur. Phys. J. B 93, 30 (2020). https://doi.org/10.1140/epjb/e2020-100595-0