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Metastability in the stochastic nearest-neighbor Kuramoto model of coupled phase oscillators
IACAPAP ArXiv (Online)
19 Dec 2024
Abstract
The Kuramoto model (KM) of $n$ coupled phase-oscillators is analyzed in this
work. The KM on a Cayley graph possesses a family of steady state solutions
called twisted states. Topologically distinct twisted states are distinguished
by the winding number $q\in\mathbb{Z}$. These states are known to be stable for
small enough $q$. In the presence of small noise, the KM exhibits metastable
transitions between $q$-twisted states: A typical trajectory remains in the
basin of attraction of a given $q$-twisted state for an exponentially long
time, but eventually transitions to the vicinity of another such state. In the
course of this transition, it passes in close proximity of a saddle of Morse
index $1$, called a relevant saddle. In this work, we provide an exhaustive
analysis of metastable transitions in the stochastic KM with nearest-neighbor
coupling. We start by analyzing the equilibria and their stability. First, we
identify all equilibria in this model. Using the discrete Fourier transform and
eigenvalue estimates for rank-1 perturbations of symmetric matrices, we
classify the equilibria by their Morse indices. In particular, we identify all
stable equilibria and all relevant saddles involved in the metastable
transitions. Further, we use Freidlin-Wentzell theory and the
potential-theoretic approach to metastability to establish the metastable
hierarchy and sharp estimates of Eyring-Kramers type for the transition times.
The former determines the precise order, in which the metastable transitions
occur, while the latter characterizes the times between successive transitions.
The theoretical estimates are complemented by numerical simulations and a
careful numerical verification of the transition times. Finally, we discuss the
implications of this work for the KM with other coupling types including
nonlocal coupling and the continuum limit as $n$ tends to infinity.
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Details
- Title
- Metastability in the stochastic nearest-neighbor Kuramoto model of coupled phase oscillators
- Creators
- Nils Berglund - Laboratoire de Mathématiques Analyse, Probabilités, Modélisation OrléansGeorgi S MedvedevGideon Simpson - Drexel University
- Publication Details
- IACAPAP ArXiv (Online)
- Resource Type
- Preprint
- Language
- English
- Academic Unit
- Mathematics
- Other Identifier
- 991022005651604721