18,253 research outputs found
Onset of synchronization in networks of second-order Kuramoto oscillators with delayed coupling: Exact results and application to phase-locked loops
We consider the inertial Kuramoto model of globally coupled oscillators
characterized by both their phase and angular velocity, in which there is a
time delay in the interaction between the oscillators. Besides the academic
interest, we show that the model can be related to a network of phase-locked
loops widely used in electronic circuits for generating a stable frequency at
multiples of an input frequency. We study the model for a generic choice of the
natural frequency distribution of the oscillators, to elucidate how a
synchronized phase bifurcates from an incoherent phase as the coupling constant
between the oscillators is tuned. We show that in contrast to the case with no
delay, here the system in the stationary state may exhibit either a subcritical
or a supercritical bifurcation between a synchronized and an incoherent phase,
which is dictated by the value of the delay present in the interaction and the
precise value of inertia of the oscillators. Our theoretical analysis,
performed in the limit , is based on an unstable manifold
expansion in the vicinity of the bifurcation, which we apply to the kinetic
equation satisfied by the single-oscillator distribution function. We check our
results by performing direct numerical integration of the dynamics for large
, and highlight the subtleties arising from having a finite number of
oscillators.Comment: 15 pages, 4 figures; v2: 16 pages, 5 figures, published versio
Canard explosion in delayed equations with multiple timescales
We analyze canard explosions in delayed differential equations with a
one-dimensional slow manifold. This study is applied to explore the dynamics of
the van der Pol slow-fast system with delayed self-coupling. In the absence of
delays, this system provides a canonical example of a canard explosion. We show
that as the delay is increased a family of `classical' canard explosions ends
as a Bogdanov-Takens bifurcation occurs at the folds points of the S-shaped
critical manifold.Comment: arXiv admin note: substantial text overlap with arXiv:1404.584
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