Date | May 14, 2021 |
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Speaker | 강명주 |

Dept. | 서울대학교 |

Room | 27-220 |

Time | 14:30-15:30 |

In this thesis, we study various Winfree-type dynamics. First, we study emergent dynamics of the continuous Winfree model with inertia and its discrete analogue. We provide sufficient conditions for the complete oscillator death to the Winfree model in the presence of inertia and the discrete-time analogue with or without inertia. We also present a uniform-in-time convergence from the discrete model to the continuous model for zero inertia case, as the time-step tends to zero. In addition, we study the emergence of asymptotic patterns in Winfree ensemble such as the partial / complete phase-locking and bump states under the effect of heterogeneous frustrations. In particular, we provide a rigorous result on the existence of bump states in a homogeneous ensemble with the same natural frequency. Moreover, we propose a Winfree type model and its mean-field limit describing the aggregation of particles on the surface of an infinite cylinder. For the proposed model, we present a sufficient framework leading to the complete oscillator death and uniform stability in a large coupling regime. We also derive the corresponding kinetic model via uniform-in-time mean-field limit. Furthermore, we study a uniform-in-time continuum limit of the lattice Winfree model and its asymptotic dynamics. For bounded measurable initial phase field, we establish a global well-posedness of classical solutions to the continuum Winfree model under suitable assumptions on coupling function, and we also show that a classical solution to the continuum Winfree model can be obtained as a limit of a sequence of lattice solutions in suitable sense.

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