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Combinatorial and probabilistic aspects of coupled oscillators
Yu, Han Baek

2018, Doctor of Philosophy, Ohio State University, Mathematics.
A prime example of complex systems is a population of \textit{coupled oscillators}, which is a network of oscillatory units with local tendency to synchronize with their neighbors. Oscillators may evolve in their oscillation cycle automatically (e.g., circadian pacemaker cells and blinking fireflies) or by excitation from neighbors (e.g., neurons in the brain and B-Z chemical oscillator). In this thesis, we propose and analyze various mathematical models of coupled oscillators in combinatorial and probabilistic perspectives.


In Part I, we introduce the $\kappa$-color \textit{firefly cellular automata} (FCA), which is a model for $\kappa$-state pulse-coupled oscillators defined for each integer parameter $\kappa\ge 3$, and obtain various theorems guaranteeing global synchronization. Our rigorous analysis on the model is based on classifying local limit cycles with their enforced dynamics, and recursively reducing global dynamics on proper subgraphs. We generalize this technique to a continuum version of the 4-color FCA, and derive global convergence on arbitrary finite tree. As an application, we obtain a memory-efficient distributed clock synchronization algorithm by composing our coupling with a spanning tree algorithm. Our composite algorithm especially suitable for synchronizing modern wireless sensor networks.


In Part II, we study the FCA on the one-dimensional infinite integer lattice $\mathbb{Z}$ as well as the 3-color cyclic cellular automaton and the Greenberg-Hastings model on arbitrary graphs in a probabilistic framework. A guiding principle of our analysis on these discrete models is to lift the dynamics to an integer-valued monotone process defined on the universal cover of underlying graph. While incorporating various techniques from probability theory such as annihilating particle systems, persistence of random walks, generating functions, large deviations for tree-indexed random walks, and mass transport principle, we also establish sharp asymptotics for persistence of Markov additive functionals.
David Sivakoff, Ph.D (Advisor)
Matthew Kahle, Ph.D (Committee Member)
Hoi Nguyen, Ph.D (Committee Member)
Elliot Paquette, Ph.D (Committee Member)
231 p.

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Yu, H. (2018). Combinatorial and probabilistic aspects of coupled oscillators . (Electronic Thesis or Dissertation). Retrieved from https://etd.ohiolink.edu/

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Yu, Han Baek. "Combinatorial and probabilistic aspects of coupled oscillators ." Electronic Thesis or Dissertation. Ohio State University, 2018. OhioLINK Electronic Theses and Dissertations Center. 19 Sep 2018.

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Yu, Han Baek "Combinatorial and probabilistic aspects of coupled oscillators ." Electronic Thesis or Dissertation. Ohio State University, 2018. https://etd.ohiolink.edu/

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