9,043 research outputs found
Analytical Phase Reduction for Weakly Nonlinear Oscillators
Phase reduction is a dimensionality reduction scheme to describe the dynamics
of nonlinear oscillators with a single phase variable. While it is crucial in
synchronization analysis of coupled oscillators, analytical results are limited
to few systems. In this letter, we analytically perform phase reduction for a
wide class of oscillators by extending the Poincar\'e-Lindstedt perturbation
theory. We exemplify the utility of our approach by analyzing an ensemble of
Van der Pol oscillators, where the derived phase model provides analytical
predictions of their collective synchronization dynamicsComment: 9 pages, 5 figure
Low-dimensional behavior of Kuramoto model with inertia in complex networks
Low-dimensional behavior of large systems of globally coupled oscillators has
been intensively investigated since the introduction of the Ott-Antonsen
ansatz. In this report, we generalize the Ott-Antonsen ansatz to second-order
Kuramoto models in complex networks. With an additional inertia term, we find a
low-dimensional behavior similar to the first-order Kuramoto model, derive a
self-consistent equation and seek the time-dependent derivation of the order
parameter. Numerical simulations are also conducted to verify our analytical
results.Comment: 6 figure
Synchronization framework for modeling transition to thermoacoustic instability in laminar combustors
We, herein, present a new model based on the framework of synchronization to
describe a thermoacoustic system and capture the multiple bifurcations that
such a system undergoes. Instead of applying flame describing function to
depict the unsteady heat release rate as the flame's response to acoustic
perturbation, the new model considers the acoustic field and the unsteady heat
release rate as a pair of nonlinearly coupled damped oscillators. By varying
the coupling strength, multiple dynamical behaviors, including limit cycle
oscillation, quasi-periodic oscillation, strange nonchaos, and chaos can be
captured. Furthermore, the model was able to qualitatively replicate the
different behaviors of a laminar thermoacoustic system observed in experiments
by Kabiraj et al.~[Chaos 22, 023129 (2012)]. By analyzing the temporal
variation of the phase difference between heat release rate oscillations and
pressure oscillations under different dynamical states, we show that the
characteristics of the dynamical states depend on the nature of synchronization
between the two signals, which is consistent with previous experimental
findings.Comment: 18 pages, 7 figure
Synchronization of heterogeneous oscillators under network modifications: Perturbation and optimization of the synchrony alignment function
Synchronization is central to many complex systems in engineering physics
(e.g., the power-grid, Josephson junction circuits, and electro-chemical
oscillators) and biology (e.g., neuronal, circadian, and cardiac rhythms).
Despite these widespread applications---for which proper functionality depends
sensitively on the extent of synchronization---there remains a lack of
understanding for how systems evolve and adapt to enhance or inhibit
synchronization. We study how network modifications affect the synchronization
properties of network-coupled dynamical systems that have heterogeneous node
dynamics (e.g., phase oscillators with non-identical frequencies), which is
often the case for real-world systems. Our approach relies on a synchrony
alignment function (SAF) that quantifies the interplay between heterogeneity of
the network and of the oscillators and provides an objective measure for a
system's ability to synchronize. We conduct a spectral perturbation analysis of
the SAF for structural network modifications including the addition and removal
of edges, which subsequently ranks the edges according to their importance to
synchronization. Based on this analysis, we develop gradient-descent algorithms
to efficiently solve optimization problems that aim to maximize phase
synchronization via network modifications. We support these and other results
with numerical experiments.Comment: 25 pages, 6 figure
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