Synchronization and Collective Nonlinear Dynamics in Complexified Oscillator Networks
Synchronization constitutes a ubiquitous phenomenon of nonlinear network dynamics and plays an essential role across natural and human-made systems. The Kuramoto model of coupled phase-oscillators constitutes a paradigmatic model for synchronization processes, with many applications in physics, biology and engineering. In the continuum limit, analytical statements about the Kuramoto model have led to a concise theory of the synchronization transition. However, finite-N systems have so far largely evaded analytic access, limiting a comprehensive understanding.
Here we propose to investigate the collective dynamics of the Kuramoto model with their traditionally real state variables analytically continued to be complex. In the past, complexification by analytic continuation has repeatedly been catalyzing major progress in our understanding the nonlinear dynamics across system classes. For instance, it has enabled an analytic theory of phase transitions, of fractal structures and PT symmetric quantum mechanics. The main direction of the proposed project is to, first, uncover and explain novel phenomena in a fundamental class of complexified coupled and networked dynamical units, with a focus on the Kuramoto model, and second, to exploit those insights to specifically understand ordering phenomena in the finite-N real-variable Kuramoto model that in turn underlies many applications. Our preliminary analysis has revealed the existence of generalized forms complex locked states that persist also for weak coupling. Moreover, numerical results indicate that the stability of those complex locked states informs us about the existence of frequency-locked sub-populations in the real model, with the imaginary parts informing us about which units belong to those subpopulations. In the proposed project, we plan to address three main classes of questions contributing to and consolidating our understanding of key overarching principles underlying such generalized forms of locking, with a focus on finite-N systems. First, which mechanisms link the stability of the complex locked states to the existence or non-existence of frequency-locked sub-populations in the original, real model? Second, how we evaluate an appropriate order parameter to learn about the degree of coordination in complexified Kuramoto networks? Third, which new types of collective dynamics emerge in classes of complexified networks also beyond the basic Kuramoto model? What are the mechanisms underlying them?
A successful project would not only connect the complex locked states to both traditionally locked and traditionally unlocked states in the original, real-variable Kuramoto model for finite-N, and thereby link to several applications. It would also reveal novel mechanisms and expand our perspective on understanding coordination phenomena via analytic continuation methods for coupled multi-dimensional dynamical systems in general.
|Deutsche Forschungsgemeinschaft (DFG, German Research Foundation)
DFG Programme Research Grants
|04/24 - 03/27
|Marc Timme, email@example.com