Diagnostics of the Synchronization of Self-Oscillatory Systems by an External Force with Varying Frequency with the Use of Wavelet Analysis

A diagnostics method based on a continuous wavelet transform is proposed. This method makes it possible to diagnose the presence of synchronization of the oscillations of a self-excited oscillator locked by an external force with a linearly modulated frequency and to distinguish such a situation from the case when an external signal leaks into self-oscillations; i.e., the signals are summed without a change in the self-oscillation frequency. The method’s efficiency is shown with the use of a Van der Pol generator and experimental physiological data as examples.

💡 Research Summary

The paper introduces a novel diagnostic technique based on the continuous wavelet transform (CWT) to determine whether a self‑excited oscillator is truly synchronized by an external force whose frequency varies linearly in time, and to distinguish this situation from a simple superposition (leakage) of the external signal with the autonomous oscillation. The authors adopt the Morlet mother wavelet, setting the central frequency ω₀ = 2π so that the wavelet scale s is the reciprocal of the Fourier frequency (s = 1/f). For a given time series x(t) the CWT yields complex coefficients W(s,t) = |W(s,t)| e^{jφₛ(t)}; the magnitude |W| reflects the energy present at scale s and time t, while the phase φₛ(t) provides a natural definition of instantaneous phase for each scale.

Synchronization is re‑defined in terms of “time‑scale synchronization”: if for a continuous range of scales