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Description
The Kuramoto model is an archetypal model for studying synchronization in groups
of nonidentical oscillators where oscillators are imbued with their own frequency and
coupled with other oscillators though a network of interactions. As the coupling
strength increases, there is a bifurcation to complete synchronization where all oscillators
move with the same frequency and show a collective rhythm. Kuramoto-like
dynamics are considered a relevant model for instabilities of the AC-power grid which
operates in synchrony under standard conditions but exhibits, in a state of failure,
segmentation of the grid into desynchronized clusters.
In this dissertation the minimum coupling strength required to ensure total frequency
synchronization in a Kuramoto system, called the critical coupling, is investigated.
For coupling strength below the critical coupling, clusters of oscillators form
where oscillators within a cluster are on average oscillating with the same long-term
frequency. A unified order parameter based approach is developed to create approximations
of the critical coupling. Some of the new approximations provide strict lower
bounds for the critical coupling. In addition, these approximations allow for predictions
of the partially synchronized clusters that emerge in the bifurcation from the
synchronized state.
Merging the order parameter approach with graph theoretical concepts leads to a
characterization of this bifurcation as a weighted graph partitioning problem on an
arbitrary networks which then leads to an optimization problem that can efficiently
estimate the partially synchronized clusters. Numerical experiments on random Kuramoto
systems show the high accuracy of these methods. An interpretation of the
methods in the context of power systems is provided.
of nonidentical oscillators where oscillators are imbued with their own frequency and
coupled with other oscillators though a network of interactions. As the coupling
strength increases, there is a bifurcation to complete synchronization where all oscillators
move with the same frequency and show a collective rhythm. Kuramoto-like
dynamics are considered a relevant model for instabilities of the AC-power grid which
operates in synchrony under standard conditions but exhibits, in a state of failure,
segmentation of the grid into desynchronized clusters.
In this dissertation the minimum coupling strength required to ensure total frequency
synchronization in a Kuramoto system, called the critical coupling, is investigated.
For coupling strength below the critical coupling, clusters of oscillators form
where oscillators within a cluster are on average oscillating with the same long-term
frequency. A unified order parameter based approach is developed to create approximations
of the critical coupling. Some of the new approximations provide strict lower
bounds for the critical coupling. In addition, these approximations allow for predictions
of the partially synchronized clusters that emerge in the bifurcation from the
synchronized state.
Merging the order parameter approach with graph theoretical concepts leads to a
characterization of this bifurcation as a weighted graph partitioning problem on an
arbitrary networks which then leads to an optimization problem that can efficiently
estimate the partially synchronized clusters. Numerical experiments on random Kuramoto
systems show the high accuracy of these methods. An interpretation of the
methods in the context of power systems is provided.
ContributorsGilg, Brady (Author) / Armbruster, Dieter (Thesis advisor) / Mittelmann, Hans (Committee member) / Scaglione, Anna (Committee member) / Strogatz, Steven (Committee member) / Welfert, Bruno (Committee member) / Arizona State University (Publisher)
Created2018
Description
The dynamics of a fluid flow inside 2D square and 3D cubic cavities
under various configurations were simulated and analyzed using a
spectral code I developed.
This code was validated against known studies in the 3D lid-driven
cavity. It was then used to explore the various dynamical behaviors
close to the onset of instability of the steady-state flow, and explain
in the process the mechanism underlying an intermittent bursting
previously observed. A fairly complete bifurcation picture emerged,
using a combination of computational tools such as selective
frequency damping, edge-state tracking and subspace restriction.
The code was then used to investigate the flow in a 2D square cavity
under stable temperature stratification, an idealized version of a lake
with warmer water at the surface compared to the bottom. The governing
equations are the Navier-Stokes equations under the Boussinesq approximation.
Simulations were done over a wide range of parameters of the problem quantifying
the driving velocity at the top (e.g. wind) and the strength of the stratification.
Particular attention was paid to the mechanisms associated with the onset of
instability of the base steady state, and the complex nontrivial dynamics
occurring beyond onset, where the presence of multiple states leads to a
rich spectrum of states, including homoclinic and heteroclinic chaos.
A third configuration investigates the flow dynamics of a fluid in a rapidly
rotating cube subjected to small amplitude modulations. The responses were
quantified by the global helicity and energy measures, and various peak
responses associated to resonances with intrinsic eigenmodes of the cavity
and/or internal retracing beams were clearly identified for the first time.
A novel approach to compute the eigenmodes is also described, making accessible
a whole catalog of these with various properties and dynamics. When the small
amplitude modulation does not align with the rotation axis (precession) we show
that a new set of eigenmodes are primarily excited as the angular velocity
increases, while triadic resonances may occur once the nonlinear regime kicks in.
under various configurations were simulated and analyzed using a
spectral code I developed.
This code was validated against known studies in the 3D lid-driven
cavity. It was then used to explore the various dynamical behaviors
close to the onset of instability of the steady-state flow, and explain
in the process the mechanism underlying an intermittent bursting
previously observed. A fairly complete bifurcation picture emerged,
using a combination of computational tools such as selective
frequency damping, edge-state tracking and subspace restriction.
The code was then used to investigate the flow in a 2D square cavity
under stable temperature stratification, an idealized version of a lake
with warmer water at the surface compared to the bottom. The governing
equations are the Navier-Stokes equations under the Boussinesq approximation.
Simulations were done over a wide range of parameters of the problem quantifying
the driving velocity at the top (e.g. wind) and the strength of the stratification.
Particular attention was paid to the mechanisms associated with the onset of
instability of the base steady state, and the complex nontrivial dynamics
occurring beyond onset, where the presence of multiple states leads to a
rich spectrum of states, including homoclinic and heteroclinic chaos.
A third configuration investigates the flow dynamics of a fluid in a rapidly
rotating cube subjected to small amplitude modulations. The responses were
quantified by the global helicity and energy measures, and various peak
responses associated to resonances with intrinsic eigenmodes of the cavity
and/or internal retracing beams were clearly identified for the first time.
A novel approach to compute the eigenmodes is also described, making accessible
a whole catalog of these with various properties and dynamics. When the small
amplitude modulation does not align with the rotation axis (precession) we show
that a new set of eigenmodes are primarily excited as the angular velocity
increases, while triadic resonances may occur once the nonlinear regime kicks in.
ContributorsWu, Ke (Author) / Lopez, Juan (Thesis advisor) / Welfert, Bruno (Thesis advisor) / Tang, Wenbo (Committee member) / Platte, Rodrigo (Committee member) / Herrmann, Marcus (Committee member) / Arizona State University (Publisher)
Created2019