Harmonic Maps on Post-Critically Finite Fractals with Applications to the Kuramoto Model

Joint work with Matthew Mizuhara (The College of New Jersey)

In this project, we study harmonic maps from fractals to the circle. Our work is motivated by the analysis of the Kuramoto model of coupled phase oscillators on graphs approximating fractals and forms part of a broader research program investigating how hierarchical network organization influences dynamics.

Hierarchical structure across multiple scales is a defining feature of many real-world networks, including synaptic connectivity in the mammalian brain, artificial neural networks, and large technological systems such as the World Wide Web. Direct modeling of connectivity in such systems is typically analytically intractable. We therefore seek graph models that capture multiscale organization while remaining amenable to rigorous analysis.

In the early 2000s, Strichartz constructed harmonic maps on the Sierpinski gasket using explicit algebraic calculations based on harmonic extension, showing that a unique harmonic map exists in each homotopy class under suitable boundary conditions.

We propose an alternative geometric construction of harmonic maps from the Sierpinski gasket to the circle. Our approach uses covering spaces tailored to each homotopy class. Continuous circle-valued functions on the fractal are lifted to real-valued functions on the corresponding covering space, where the harmonic extension algorithm can be applied. The desired harmonic map is then obtained by restricting the resulting harmonic function to a fundamental domain and projecting its values back to the circle. Each covering space is constructed to reflect the intrinsic topology of a given homotopy class.

This method generalizes naturally to other post-critically finite fractals, including the 3-level Sierpinski gasket, the hexagasket, and the pentagasket. Moreover, the lifting procedure enables the use of analytic techniques that are otherwise unavailable for circle-valued maps. In subsequent work, we adapt Γ-convergence methods to study stable steady states of the Kuramoto model on graphs approximating fractals. This program provides a rigorous analytical framework for studying synchronization and pattern formation on networks with fractal or hierarchical architecture.

T-valued harmonic maps of different homotopy types (Reproduced from Medvedev & Mizuhara, arXiv:2407.16817.)

Publications

• G.S. Medvedev and M. Mizuhara, The Kuramoto Model on the Sierpinski Gasket I: Harmonic Maps from Post-Critically Finite Fractals to the Circle, arXiv.
• G.S. Medvedev and M. Mizuhara, The Kuramoto Model on the Sierpinski Gasket II: Twisted States, arXiv.

Funding

• NSF Award #2406941 , Collaborative Research: Emerging Applications of Self-Similarity in Dynamical Networks, 2024-2027.

Graphon Dynamical Systems

Many fundamental systems in physics and biology can be modeled as networks of interacting dynamical elements. Examples include neuronal networks in the brain, biological populations (e.g., swarms of fireflies), social networks, artificial neural networks, power grids, and the Internet.

Mathematical models of such systems are typically very high dimensional, structurally heterogeneous and complex. Their analysis requires the integration of methods from dynamical systems, graph theory, and probability.

A central question in the theory of interacting dynamical systems, as well as in its applications, concerns the relationship between network structure and emergent dynamics. Until recently, this question could be addressed analytically only for special classes of (typically highly symmetric) connectivity patterns.

         

Graphon representing a small-world graph.

The situation has started to change with the development of graph limit theory by László Lovász and collaborators (see L. Lovász, Large Networks and Graph Limits, AMS, 2013). The central idea of this theory is to represent convergent sequences of graphs and their limit objects by measurable functions (graphons) defined on the unit square.

Graph limit theory provides a natural and powerful analytical framework for the study of large networks. After attending László Lovász's lectures on graph limits in the early 2010s, I began to see connections with problems in dynamical networks and became interested in adapting ideas from this theory to the analysis of interacting dynamical systems.

My first two papers in this direction established a framework for the rigorous derivation and analysis of the continuum limit for dynamical models on deterministic and random dense graphs:

• G.S. Medvedev, The Nonlinear Heat Equation on Dense Graphs and Graph Limits, SIAM Journal on Mathematical Analysis, 46(4), 2743-2766 (2014). abstract   pdf
• G.S. Medvedev, The Nonlinear Heat Equation on W-Random Graphs, Archive for Rational Mechanics and Analysis, 212(3), 781-803 (2014). abstract   pdf

In subsequent work, I extended these techniques to dynamical models on sparse random graphs:

• G.S. Medvedev, The Continuum Limit of the Kuramoto Model on Sparse Random Graphs, Communications in Mathematical Sciences, 17(4), 883-898 (2019).

In joint work with Paul Dupuis, we established a Large Deviation Principle for interacting dynamical systems on random graphs:

• P. Dupuis and G.S. Medvedev, The Large Deviation Principle for Interacting Dynamical Systems on Random Graphs, Communications in Mathematical Physics, 390 (2022) (arXiv).

This result establishes a Large Deviation Principle for sequences of W-random graphs, substantially generalizing earlier work of Chatterjee and Varadhan on Erdos-Renyi graphs.

In joint work with Dmitry Kalyuzhnyi-Verbovetskyi, we derived and rigorously justified the mean-field limit for interacting dynamical systems with random parameters on graphs. In particular, we extended Neunzert's method for proving the validity of the Vlasov equation to the non-exchangeable setting of coupled dynamical systems on graphs. We used the Kuramoto model of coupled phase oscillators as a convenient setting to present our approach. However, the key ideas used to address non-exchangeability - notably the use of local empirical measures, naturally apply to other dynamical models on graphs.

• D. Kalyuzhnyi-Verbovetskyi and G.S. Medvedev, The Mean Field Equation for the Kuramoto Model on Graph Sequences with Non-Lipschitz Limit, SIAM J. Math. Anal., 50 (2018), no. 3, 2441-2465. abstract   pdf

Funding