Low-Dimensional Behavior of a Kuramoto Model with Inertia and Hebbian Learning
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Publication:6394441
DOI10.1063/5.0092378arXiv2203.12090OpenAlexW4389615447MaRDI QIDQ6394441
Tachin Ruangkriengsin, Mason A. Porter
Publication date: 22 March 2022
Abstract: We study low-dimensional dynamics in a Kuramoto model with inertia and Hebbian learning. In this model, the coupling strength between oscillators depends on the phase differences between the oscillators and changes according to a Hebbian learning rule. We analyze the special case of two coupled oscillators, which yields a five-dimensional dynamical system that decouples into a two-dimensional longitudinal system and a three-dimensional transverse system. We readily write an exact solution of the longitudinal system, and we then focus our attention on the transverse system. We classify the stability of the transverse system's equilibrium points using linear stability analysis. We show that the transverse system is dissipative and that all of its trajectories are eventually confined to a bounded region. We approximate the transverse system's limiting behavior and demarcate the parameter regions of three qualitatively different behaviors. Using insights from our analysis of the low-dimensional dynamics, we study the original high-dimensional system in a situation in which we draw the intrinsic frequencies of the oscillators from Gaussian distributions with different variances.
Full work available at URL: https://doi.org/10.1063/5.0092378
Stability theory for ordinary differential equations (34Dxx) Qualitative theory for ordinary differential equations (34Cxx) Mathematical biology in general (92Bxx)
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