Flexible, Automated Determination of Incommensurable Basis Frequencies Underlying Quasi-Periodic Processes

In quasi-periodic processes, where power spectra consist of discrete “observed” frequencies with at least one pair incommensurable (i.e., having an irrational ratio), determining the underlying “basis” frequencies is of interest in understanding dynamical behavior. Typically, one seeks, by trial and error, a set of basis frequencies (frequently two) whose integer multiples can be summed to reconstruct the observed frequencies. This can be daunting if there are more than a few observed frequencies. Here we describe an automated method that treats basis frequencies as positive reals and reconstruction coefficients as integers and uses a mixed integer nonlinear programming approach to find the combination minimizing the sum of the magnitudes of the integers, which is in some sense the simplest solution. The method is quite efficient even with many observed frequencies, and can accommodate extrinsic information about either observed frequency precision or one or more underlying frequencies. Compared to alternatives, this method provides greater accuracy and gives a posteriori information on accuracy of observed frequencies. We demonstrate the approach for quasi-periodic experimental buoyancy-driven convection data of Fein, Heutmaker & Gollub (1985) and cylinder wake data of Van Atta & Gharib (1987), and computational data of Blanchard et al. (2019).