Universal description of stochastic oscillators - higher dimensional examples, extraction of the mapping from data, and networks of oscillators

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Many important non-equilibrium systems display oscillatory behavior that is shaped by pronounced fluctuations in phase and amplitude. Different mathematical models with distinct mechanisms have been suggested for such stochastic oscillators: (i) dynamics with a focus fixed point endowed with noise (the damped harmonic oscillator with thermal fluctuations would be a prominent example); (ii) limit cycles and relaxation oscillators perturbed by white noise; (iii) excitable systems driven by fluctuations; (iv) heteroclinic systems driven by noise. Most of these models are strongly nonlinear and are thus difficult to treat theoretically. A recent theory, put forward by my collaborators and me, provides a universal framework for the description of stochastic oscillators in terms of one of the eigenfunctions of the backward Kolmogorov operator. After projecting the system's variables to a new complex-valued variable, the spontaneous fluctuation statistics, the linear response of the oscillator, and its cross-correlation with another stochastic oscillator to which it is coupled, are all captured by simple and exact analytical expressions. This description is independent of the mechanism for the stochastic oscillation. In this project I want to apply and generalize this theory in three respects: (i) it should be applied to oscillators that have higher dimensions than two going beyond what has been studied before; (ii) the nonlinear transformation at the core of the procedure should be extracted from (simulation or experimental) data; (iii) the theory for coupled stochastic oscillators will be more thoroughly grounded and, moreover, extended to the case of networks of stochastic oscillators (more than two oscillators, the case previously investigated). These explorations and extensions of the framework have potential applications for the theoretical description of many systems outside of thermodynamic equilibrium, in particular in biology.