3,475 research outputs found
Mean field theory of assortative networks of phase oscillators
Employing the Kuramoto model as an illustrative example, we show how the use
of the mean field approximation can be applied to large networks of phase
oscillators with assortativity. We then use the ansatz of Ott and Antonsen
[Chaos 19, 037113 (2008)] to reduce the mean field kinetic equations to a
system of ordinary differential equations. The resulting formulation is
illustrated by application to a network Kuramoto problem with degree
assortativity and correlation between the node degrees and the natural
oscillation frequencies. Good agreement is found between the solutions of the
reduced set of ordinary differential equations obtained from our theory and
full simulations of the system. These results highlight the ability of our
method to capture all the phase transitions (bifurcations) and system
attractors. One interesting result is that degree assortativity can induce
transitions from a steady macroscopic state to a temporally oscillating
macroscopic state through both (presumed) Hopf and SNIPER (saddle-node,
infinite period) bifurcations. Possible use of these techniques to a broad
class of phase oscillator network problems is discussed.Comment: 8 pages, 7 figure
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
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