Detecting Dynamical States from Noisy Time Series using Bicoherence

Abstract: Deriving meaningful information from observational data is often restricted by many limiting factors, the most important of which is the presence of noise. In this work, we present the use of the bicoherence function to extract information about the underlying nonlinearity from noisy time series. We show that a system evolving in the presence of noise which has its dynamical state concealed from quantifiers like the power spectrum and correlation dimension D2, can be revealed using the bicoherence function. We define an index called main peak bicoherence function as the bicoherence associated with the maximal power spectral peak. We show that this index is extremely useful while dealing with quasi-periodic data as it can distinguish strange non chaos from quasi periodicity even with added noise. We demonstrate this in a real world scenario, by taking the bicoherence of variable stars showing period doubling and strange non-chaotic behavior. Our results indicate that bicoherence analysis can also bypass the method of surrogate analysis using Fourier phase randomization, used to differentiate linear stochastic processes from non linear ones, in conventional methods involving measures like D2.