  Many real-world systems modeled using differential equations involve unknown
or uncertain parameters. Standard approaches to address parameter estimation
inverse problems in this setting typically focus on estimating constants; yet
some unobservable system parameters may vary with time without known evolution
models. In this work, we propose a novel approximation method inspired by the
Fourier series to estimate time-varying parameters in deterministic dynamical
systems modeled with ordinary differential equations. Using ensemble Kalman
filtering in conjunction with Fourier series-based approximation models, we
detail two possible implementation schemes for sequentially updating the
time-varying parameter estimates given noisy observations of the system states.
We demonstrate the capabilities of the proposed approach in estimating periodic
parameters, both when the period is known and unknown, as well as non-periodic
time-varying parameters of different forms with several computed examples using
a forced harmonic oscillator. Results emphasize the importance of the
frequencies and number of approximation model terms on the time-varying
parameter estimates and corresponding dynamical system predictions.
