A practical method for estimating coupling functions in complex dynamical systems

A foremost challenge in modern network science is the inverse problem of reconstruction (inference) of coupling equations and network topology from the measurements of the network’s dynamics. Of particular interest are the methods that can operate on real (empirical) data without interfering with the system. One of such earlier attempts (Tokuda et al. 2007 Phys. Rev. Lett. 99, 064101) was a method suited for general limit-cycle oscillators, yielding both oscillators’ natural frequencies and coupling functions between them (phase equations) from empirically measured time series. The present paper reviews the above method in a way comprehensive to domain-scientists other than physics. It also presents applications of the method to (i) detection of the network connectivity, (ii) inference of the phase sensitivity function, (iii) approximation of the interaction among phase-coherent chaotic oscillators, and (iv) experimental data from a forced Van der Pol electric circuit. This reaffirms th..., External forcing was injected from a function generator (Keysight 33500B) to Van der Pol circuit.

To obtain the phase sensitivity function, impulses (stimulus duration: 380 µs, stimulus strength: 3 V) were randomly injected as the external force. The circuit output as well as the input impulses were simultaneously measured.

The first file (PRC.csv) contains phase sensitivity function (also called ``infinitesimal phase response curve''). The amount of phase shift induced by external stimulus is described as a function of the timing at which the stimulus was injected. As the external force, impulses (stimulus duration: 380 µs, stimulus strength: 3 V) were randomly injected. Natural frequency of the oscillator circuit measured before the stimulus experiment was 110.5 Hz.

The second file (106hz100m100m1p5v12p5ksmpl.csv) contains Van der Pol circuit dynamics with a sinusoidal forcing (forcing frequency: 106 Hz). The forcing amplitude was varied from 0 V to 1.5 V. The circuit output ...,