609 research outputs found
Low Dimensional Dynamics of the Kuramoto Model with Rational Frequency Distributions
The Kuramoto model is a paradigmatic tool for studying the dynamics of
collective behavior in large ensembles of coupled dynamical systems. Over the
past decade a great deal of progress has been made in analytical descriptions
of the macroscopic dynamics of the Kuramoto mode, facilitated by the discovery
of Ott and Antonsen's dimensionality reduction method. However, the vast
majority of these works relies on a critical assumption where the oscillators'
natural frequencies are drawn from a Cauchy, or Lorentzian, distribution, which
allows for a convenient closure of the evolution equations from the
dimensionality reduction. In this paper we investigate the low dimensional
dynamics that emerge from a broader family of natural frequency distributions,
in particular a family of rational distribution functions. We show that, as the
polynomials that characterize the frequency distribution increase in order, the
low dimensional evolution equations become more complicated, but nonetheless
the system dynamics remain simple, displaying a transition from incoherence to
partial synchronization at a critical coupling strength. Using the low
dimensional equations we analytically calculate the critical coupling strength
corresponding to the onset of synchronization and investigate the scaling
properties of the order parameter near the onset of synchronization. These
results agree with calculations from Kuramoto's original self-consistency
framework, but we emphasize that the low dimensional equations approach used
here allows for a true stability analysis categorizing the bifurcations
Control of coupled oscillator networks with application to microgrid technologies
The control of complex systems and network-coupled dynamical systems is a
topic of vital theoretical importance in mathematics and physics with a wide
range of applications in engineering and various other sciences. Motivated by
recent research into smart grid technologies we study here control of
synchronization and consider the important case of networks of coupled phase
oscillators with nonlinear interactions--a paradigmatic example that has guided
our understanding of self-organization for decades. We develop a method for
control based on identifying and stabilizing problematic oscillators, resulting
in a stable spectrum of eigenvalues, and in turn a linearly stable synchronized
state. Interestingly, the amount of control, i.e., number of oscillators,
required to stabilize the network is primarily dictated by the coupling
strength, dynamical heterogeneity, and mean degree of the network, and depends
little on the structural heterogeneity of the network itself
Coexisting chaotic and multi-periodic dynamics in a model of cardiac alternans
The spatiotemporal dynamics of cardiac tissue is an active area of research
for biologists, physicists, and mathematicians. Of particular interest is the
study of period-doubling bifurcations and chaos due to their link with cardiac
arrhythmogenesis. In this paper we study the spatiotemporal dynamics of a
recently developed model for calcium-driven alternans in a one dimensional
cable of tissue. In particular, we observe in the cable coexistence of regions
with chaotic and multi-periodic dynamics over wide ranges of parameters. We
study these dynamics using global and local Lyapunov exponents and spatial
trajectory correlations. Interestingly, near nodes -- or phase reversals --
low-periodic dynamics prevail, while away from the nodes the dynamics tend to
be higher-periodic and eventually chaotic. Finally, we show that similar
coexisting multi-periodic and chaotic dynamics can also be observed in a
detailed ionic model
Spectral properties of the hierarchical product of graphs
The hierarchical product of two graphs represents a natural way to build a
larger graph out of two smaller graphs with less regular and therefore more
heterogeneous structure than the Cartesian product. Here we study the
eigenvalue spectrum of the adjacency matrix of the hierarchical product of two
graphs. Introducing a coupling parameter describing the relative contribution
of each of the two smaller graphs, we perform an asymptotic analysis for the
full spectrum of eigenvalues of the adjacency matrix of the hierarchical
product. Specifically, we derive the exact limit points for each eigenvalue in
the limits of small and large coupling, as well as the leading-order relaxation
to these values in terms of the eigenvalues and eigenvectors of the two smaller
graphs. Given its central roll in the structural and dynamical properties of
networks, we study in detail the Perron-Frobenius, or largest, eigenvalue.
Finally, as an example application we use our theory to predict the epidemic
threshold of the Susceptible-Infected-Susceptible model on a hierarchical
product of two graphs
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