| Abstract: |
| Coupled phase oscillators provide a prototypical setting for studying spontaneous synchronization and other pattern formation. The interplay between network topology and emergent phenomena remains a central theme in their study. Some real-world networks, such as coupled neurons and the Internet, exhibit self-similarity at multiple scales, so there is a need to analyze coupled oscillators on hierarchical/fractal structures. To that end, we study Kuramoto oscillators on self-similar networks approximating fractals. We show that the complex topology of fractals gives rise to a rich diversity of equilibria, generalizing well-studied twisted states found on simpler networks. We introduce an approach for constructing and classifying these states by combining tools from fractal geometry, topology, and analysis. Our method relies on harmonic extensions, which provide exact solutions to the Laplace equation on the fractal, and uses T-convergence to establish convergence of the nonlinear problems. This work is in collaboration with Georgi Medvedev and is supported by the U.S. National Science Foundation. |
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