Steady State Statistics of Emergent Patterns in a Ring of Oscillators

Abstract

Networks of coupled nonlinear oscillators model a broad class of physical, chemical and biological systems. Understanding emergent patterns in such networks is an ongoing effort with profound implications for different fields. In this work, we analytically and numerically study a symmetric ring of N coupled self-oscillators of Van der Pol type under external stochastic forcing. The system is proposed as a model of the thermo- and aeroacoustic interactions of sound fields in rigid enclosures with compact source regions in a can-annular combustor. The oscillators are connected via linear resistive coupling with nonlinear saturation. After transforming the system to amplitude-phase coordinates, deterministic and stochastic averaging is performed to eliminate the fast oscillating terms. By projecting the potential of the slow-flow dynamics onto the phase-locked quasi-limit cycle solutions, we obtain a compact, low-order description of the (de-)synchronization transition for an arbitrary number of oscillators. The stationary probability density function of the state variables is derived from the Fokker--Planck equation, studied for varying parameter values and compared to time series simulations. We leverage our analysis to offer explanations for features of acoustic pressure spectrograms observed in real-world gas turbines.

arXiv comment16 figures

References in corpus (14)

• The Kuramoto model in complex networks
• Robustness of the noise-induced phase synchronization in a general class of limit cycle oscillators
• Computing with Networks of Nonlinear Mechanical Oscillators
• Symmetric States Requiring System Asymmetry
• Noise-Induced Synchronization of a Large Population of Globally Coupled Nonidentical Oscillators
• Noise-Induced Synchronization, Desynchronization, and Clustering in Globally Coupled Nonidentical Oscillators
• Entanglement dynamics of two coupled mechanical oscillators in modulated optomechanics
• The Implicit and the Inverse Function theorems: easy proofs
• Interplay of coupling and common noise at the transition to synchrony in oscillator populations
• Collective phase dynamics of globally coupled oscillators: Noise-induced anti-phase synchronization
• Colored noise induces synchronization of limit cycle oscillators
• Patterns of patterns of synchronization: Noise induced attractor switching in rings of coupled nonlinear oscillators
• Processing time-series of randomly forced self-oscillators: the example of beer bottle whistling
• Coupling-Induced Instability in a Ring of Thermoacoustic Oscillators