相关论文: Continuum limits of coupled oscillator networks de…
The number $\mathcal{N}$ of stable fixed points of locally coupled Kuramoto models depends on the topology of the network on which the model is defined. It has been shown that cycles in meshed networks play a crucial role in determining…
We prove that the Kuramoto model on a graph can contain infinitely many non-equivalent stable equilibria. More precisely, we prove that for every positive integer d there is a connected graph such that the set of stable equilibria contains…
We study bifurcations of twisted solutions in a continuum limit (CL) for the Kuramoto model (KM) of identical oscillators defined on nearest neighbor graphs, which may be deterministic dense, random dense or random sparse, when it may have…
The Kuramoto model of coupled phase oscillators on small-world (SW) graphs is analyzed in this work. When the number of oscillators in the network goes to infinity, the model acquires a family of steady state solutions of degree q, called…
Networks with different levels of interactions, including multilayer and multiplex networks, can display a rich diversity of dynamical behaviors and can be used to model and study a wide range of systems. Despite numerous efforts to…
We propose a Kuramoto model of coupled oscillators on a time-varying graph, whose dynamics is dictated by a Markov process in the space of graphs. The simplest representative is considering a base graph and then the subgraph determined by…
We solve a longstanding stability problem for the Kuramoto model of coupled oscillators. This system has attracted mathematical attention, in part because of its applications in fields ranging from neuroscience to condensed-matter physics,…
Using the theory of $L^p$-graphons (Borgs et al, 2014), we derive and rigorously justify the continuum limit for systems of differential equations on sparse random graphs. Specifically, we show that the solutions of the initial value…
The incoherent state of the Kuramoto model of coupled oscillators exhibits marginal modes in mean field theory. We demonstrate that corrections due to finite size effects render these modes stable in the subcritical case, i.e. when the…
We investigate algebraic and topological signatures of networks of coupled oscillators. Translating dynamics into a system of algebraic equations enables us to identify classes of network topologies that exhibit unexpected behaviors. Many…
Designing high-performing networks requires optimizing for functionality while respecting physical, geometric, or budget constraints. Yet, mathematical and computational tools to design such systems remain limited, particularly for…
Kuramoto model is one of the most prominent models for the synchronization of coupled oscillators. It has long been a research hotspot to understand how natural frequencies, the interaction between oscillators, and network topology…
The article considers systems of interacting particles on networks with adaptively coupled dynamics. Such processes appear frequently in natural processes and applications. Relying on the notion of graph convergence, we prove that for large…
We consider self-loops and multiple edges in the configuration model as the size of the graph tends to infinity. The interest in these random variables is due to the fact that the configuration model, conditioned on being simple, is a…
This paper studies the $p$-biharmonic equation on graphs, which arises in point cloud processing and can be interpreted as a natural extension of the graph $p$-Laplacian from the perspective of hypergraph. The asymptotic behavior of the…
For general networks of pulse-coupled oscillators, including regular, random, and more complex networks, we develop an exact stability analysis of synchronous states. As opposed to conventional stability analysis, here stability is…
We study a nonlocal diffusion equation approximating the dynamics of coupled phase oscillators on large graphs. Under appropriate assumptions, the model has a family of steady state solutions called twisted states. We prove a sufficient…
We investigate the dynamics of the adaptive Kuramoto model with slow adaptation in the continuum limit, $N\to\infty$. This model is distinguished by dense multistability, where multiple states coexist for the same system parameters. The…
We consider singularly perturbed convection-diffusion equations on one-dimensional networks (metric graphs) as well as the transport problems arising in the vanishing diffusion limit. Suitable coupling condition at inner vertices are…
Weakly coupled oscillators are used throughout the physical sciences, particularly in mathematical neuroscience to describe the interaction of neurons in the brain. Systems of weakly coupled oscillators have a well-known decomposition to a…