Spectra of delay-coupled heterogeneous noisy nonlinear oscillators

Nonlinear oscillators that are subject to noise and delayed interaction have been used to describe a number of dynamical phenomena in Physics and beyond. Here we study the spectral statistics (power and cross-spectral densities) of a small number of noisy nonlinear oscillators and derive analytical approximations for these spectra. In our paper, individual oscillators are described by the normal form of a supercritical or subcritical Hopf bifurcation supplemented by Gaussian white noise. Oscillators can be distinguished from each other by their frequency, bifurcation parameter, and noise intensity. Extending previous results from the literature, we first calculate in linear response theory the power spectral density and response function of the single oscillator in both super- and subcritical parameter regime and test them against numerical simulations. For small heterogeneous groups of oscillators (N = 2 or 3), which are coupled by a delayed linear term, we use a linear response ansatz to derive approximations for the power and cross-spectral densities of the oscillators within this small network. These approximations are confirmed by comparison with extensive numerical simulations. Using the theory we relate the peaks in the spectra of the homogeneous system (identical oscillators) to periodic solutions of the deterministic (noiseless) system. For two delay-coupled subcritical Hopf oscillators, we show that the coupling can enhance the coherence resonance effect, which is known to occur for the single subcritical oscillator. In the case of heterogeneous oscillators, we find that the delay-induced characteristic profile of the spectra is conserved for moderate frequency detuning.