Inference of Hopf normal form representations from oscillatory time series

We introduce a probabilistic framework for representing noisy oscillatory time series using the Hopf normal form, enabling inference of low-dimensional dynamical properties without requiring knowledge of the governing equations. By embedding the normal form in a state–space model, the method jointly infers latent states and system parameters, providing estimates of the natural frequency, Floquet exponent, and asymptotic phase in the presence of observation noise. Combined with complex Gaussian process regression, the framework further reconstructs phase and amplitude sensitivity functions from the inferred latent representation. Numerical experiments using the van der Pol and Goodwin oscillators demonstrate that the proposed approach can recover normal-form parameters and phase–amplitude structure from noisy oscillatory time series while maintaining robustness under strong observation noise. These results illustrate how normal-form theory can be integrated with probabilistic state–space inference to obtain low-dimensional representations of oscillatory dynamics from time-series observations.