56,390 research outputs found
Global analysis of a continuum model for monotone pulse-coupled oscillators
We consider a continuum of phase oscillators on the circle interacting
through an impulsive instantaneous coupling. In contrast with previous studies
on related pulse-coupled models, the stability results obtained in the
continuum limit are global. For the nonlinear transport equation governing the
evolution of the oscillators, we propose (under technical assumptions) a global
Lyapunov function which is induced by a total variation distance between
quantile densities. The monotone time evolution of the Lyapunov function
completely characterizes the dichotomic behavior of the oscillators: either the
oscillators converge in finite time to a synchronous state or they
asymptotically converge to an asynchronous state uniformly spread on the
circle. The results of the present paper apply to popular phase oscillators
models (e.g. the well-known leaky integrate-and-fire model) and draw a strong
parallel between the analysis of finite and infinite populations. In addition,
they provide a novel approach for the (global) analysis of pulse-coupled
oscillators.Comment: 33 page
Synchronization of Integrate and Fire oscillators with global coupling
In this article we study the behavior of globally coupled assemblies of a
large number of Integrate and Fire oscillators with excitatory pulse-like
interactions. On some simple models we show that the additive effects of pulses
on the state of Integrate and Fire oscillators are sufficient for the
synchronization of the relaxations of all the oscillators. This synchronization
occurs in two forms depending on the system: either the oscillators evolve ``en
bloc'' at the same phase and therefore relax together or the oscillators do not
remain in phase but their relaxations occur always in stable avalanches. We
prove that synchronization can occur independently of the convexity or
concavity of the oscillators evolution function. Furthermore the presence of
disorder, up to some level, is not only compatible with synchronization, but
removes some possible degeneracy of identical systems and allows new mechanisms
towards this state.Comment: 37 pages, 19 postscript figures, Latex 2
Phase shifts of synchronized oscillators and the systolic/diastolic blood pressure relation
We study the phase-synchronization properties of systolic and diastolic
arterial pressure in healthy subjects. We find that delays in the oscillatory
components of the time series depend on the frequency bands that are
considered, in particular we find a change of sign in the phase shift going
from the Very Low Frequency band to the High Frequency band. This behavior
should reflect a collective behavior of a system of nonlinear interacting
elementary oscillators. We prove that some models describing such systems, e.g.
the Winfree and the Kuramoto models offer a clue to this phenomenon. For these
theoretical models there is a linear relationship between phase shifts and the
difference of natural frequencies of oscillators and a change of sign in the
phase shift naturally emerges.Comment: 8 figures, 9 page
On the onset of synchronization of Kuramoto oscillators in scale-free networks
Despite the great attention devoted to the study of phase oscillators on
complex networks in the last two decades, it remains unclear whether scale-free
networks exhibit a nonzero critical coupling strength for the onset of
synchronization in the thermodynamic limit. Here, we systematically compare
predictions from the heterogeneous degree mean-field (HMF) and the quenched
mean-field (QMF) approaches to extensive numerical simulations on large
networks. We provide compelling evidence that the critical coupling vanishes as
the number of oscillators increases for scale-free networks characterized by a
power-law degree distribution with an exponent , in line
with what has been observed for other dynamical processes in such networks. For
, we show that the critical coupling remains finite, in agreement
with HMF calculations and highlight phenomenological differences between
critical properties of phase oscillators and epidemic models on scale-free
networks. Finally, we also discuss at length a key choice when studying
synchronization phenomena in complex networks, namely, how to normalize the
coupling between oscillators
Hopf normal form with symmetry and reduction to systems of nonlinearly coupled phase oscillators
Coupled oscillator models where oscillators are identical and
symmetrically coupled to all others with full permutation symmetry are
found in a variety of applications. Much, but not all, work on phase
descriptions of such systems consider the special case of pairwise coupling
between oscillators. In this paper, we show this is restrictive - and we
characterise generic multi-way interactions between oscillators that are
typically present, except at the very lowest order near a Hopf bifurcation
where the oscillations emerge. We examine a network of identical weakly coupled
dynamical systems that are close to a supercritical Hopf bifurcation by
considering two parameters, (the strength of coupling) and
(an unfolding parameter for the Hopf bifurcation). For small enough
there is an attractor that is the product of stable limit cycles; this
persists as a normally hyperbolic invariant torus for sufficiently small
. Using equivariant normal form theory, we derive a generic normal
form for a system of coupled phase oscillators with symmetry. For fixed
and taking the limit , we show that the
attracting dynamics of the system on the torus can be well approximated by a
coupled phase oscillator system that, to lowest order, is the well-known
Kuramoto-Sakaguchi system of coupled oscillators. The next order of
approximation genericlly includes terms with up to four interacting phases,
regardless of . Using a normalization that maintains nontrivial interactions
in the limit , we show that the additional terms can lead
to new phenomena in terms of coexistence of two-cluster states with the same
phase difference but different cluster size
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