Two of the main tools for quantifying the complexity of time series data are the correlation dimension and permutation entropy.
Correlation Dimension – Quantifies the self-similarity of a geometrical object, in this case a phase space trajectory that represents the state of the dynamical system. It is one of several fractal dimension measures, this being the most suited for application to experimental data. It gives a value which is consistent with typical dimension values (e.g. 1-D line, 2-D surface), however it is not particularly robust to noise.
Permutation Entropy – A measure of time series complexity based on the entropy of ordinal permutations. The ordinal pattern symbols are determined by the relative amplitude of points in the time series. It is quick and easy to implement on experimental data and is robust to noise. The limitation is that the value obtained is only relative and highly complex data can give values of similar magnitude to a stochastic process.