Data-driven inference of Hopf normal form representations from oscillatory time series

We introduce a data-driven framework that maps noisy oscillatory time series directly onto the Hopf normal form, enabling inference of underlying dynamics without knowledge of governing equations. By embedding the normal form in a probabilistic state-space model, the method jointly infers latent states and system parameters, yielding robust estimates of the natural frequency, Floquet exponent, and asymptotic phase even far from the bifurcation point and under strong noise. Combined with complex Gaussian process regression, the approach further reconstructs phase and amplitude sensitivity functions from data. Benchmarks on the van der Pol oscillator demonstrate substantially improved accuracy and noise robustness compared with existing phase-based and regression methods. This work establishes a direct bridge between normal-form theory and statistical inference, providing a general and practical route to low-dimensional descriptions of oscillatory dynamics in complex systems.