Reduced-order modeling of dynamical systems coupled with the mean field of oscillators: Application to thermoacoustic...

Dynamical systems in which some of the subsystems are modeled as a population of oscillators and interact with other subsystems through the mean field of oscillators are commonly observed in nature. In this paper, we present a framework for analyzing synchronization in such systems. While these models are conceptually simple, they quickly become computationally expensive and analytically intractable as the number of oscillators grows. To address this challenge, a macroscopic description of the oscillator population is often derived by considering the thermodynamic limit. For analytical purposes, the oscillator distribution is chosen from standard symmetric distributions, simplifying the representation, and the number of equations can be reduced to around the order of 1. However, such standard symmetric distributions are rare in practical systems, and very few works in the literature have attempted to address this problem. In this paper, we propose a generalized framework to reduce the dimensionality of a dynamical system where part of the subsystems exhibit behavior similar to a set of oscillators. The proposed method applies to nonstandard distributions observed in practical systems and uses model reduction methods, such as the Ott-Antonsen ansatz, to obtain a reduced-order model for dynamical systems. Our process reduces the high-dimensional models to around order of ten number of equations. These reduced models can then be studied analytically to identify the stability regimes of the system. We demonstrate the generalizability of our method for a thermoacoustic system, where the heat source is modeled using a set of oscillators based on the previous theoretical studies.