453 research outputs found
Slowly varying control parameters, delayed bifurcations and the stability of spikes in reaction-diffusion systems
We present three examples of delayed bifurcations for spike solutions of
reaction-diffusion systems. The delay effect results as the system passes
slowly from a stable to an unstable regime, and was previously analysed in the
context of ODE's in [P.Mandel, T.Erneux, J.Stat.Phys, 1987]. It was found that
the instability would not be fully realized until the system had entered well
into the unstable regime. The bifurcation is said to have been "delayed"
relative to the threshold value computed directly from a linear stability
analysis. In contrast, we analyze the delay effect in systems of PDE's. In
particular, for spike solutions of singularly perturbed generalized
Gierer-Meinhardt (GM) and Gray-Scott (GS) models, we analyze three examples of
delay resulting from slow passage into regimes of oscillatory and competition
instability. In the first example, for the GM model on the infinite real line,
we analyze the delay resulting from slowly tuning a control parameter through a
Hopf bifurcation. In the second example, we consider a Hopf bifurcation on a
finite one-dimensional domain. In this scenario, as opposed to the extrinsic
tuning of a system parameter through a bifurcation value, we analyze the delay
of a bifurcation triggered by slow intrinsic dynamics of the PDE system. In the
third example, we consider competition instabilities of the GS model triggered
by the extrinsic tuning of a feed rate parameter. In all cases, we find that
the system must pass well into the unstable regime before the onset of
instability is fully observed, indicating delay. We also find that delay has an
important effect on the eventual dynamics of the system in the unstable regime.
We give analytic predictions for the magnitude of the delays as obtained
through analysis of certain explicitly solvable nonlocal eigenvalue problems.
The theory is confirmed by numerical solutions of the full PDE systems.Comment: 31 pages, 20 figures, submitted to Physica D: Nonlinear Phenomen
A normal form for excitable media
We present a normal form for travelling waves in one-dimensional excitable
media in form of a differential delay equation. The normal form is built around
the well-known saddle-node bifurcation generically present in excitable media.
Finite wavelength effects are captured by a delay. The normal form describes
the behaviour of single pulses in a periodic domain and also the richer
behaviour of wave trains. The normal form exhibits a symmetry preserving Hopf
bifurcation which may coalesce with the saddle-node in a Bogdanov-Takens point,
and a symmetry breaking spatially inhomogeneous pitchfork bifurcation. We
verify the existence of these bifurcations in numerical simulations. The
parameters of the normal form are determined and its predictions are tested
against numerical simulations of partial differential equation models of
excitable media with good agreement.Comment: 22 pages, accepted for publication in Chao
Mean-field equations for stochastic firing-rate neural fields with delays: Derivation and noise-induced transitions
In this manuscript we analyze the collective behavior of mean-field limits of
large-scale, spatially extended stochastic neuronal networks with delays.
Rigorously, the asymptotic regime of such systems is characterized by a very
intricate stochastic delayed integro-differential McKean-Vlasov equation that
remain impenetrable, leaving the stochastic collective dynamics of such
networks poorly understood. In order to study these macroscopic dynamics, we
analyze networks of firing-rate neurons, i.e. with linear intrinsic dynamics
and sigmoidal interactions. In that case, we prove that the solution of the
mean-field equation is Gaussian, hence characterized by its two first moments,
and that these two quantities satisfy a set of coupled delayed
integro-differential equations. These equations are similar to usual neural
field equations, and incorporate noise levels as a parameter, allowing analysis
of noise-induced transitions. We identify through bifurcation analysis several
qualitative transitions due to noise in the mean-field limit. In particular,
stabilization of spatially homogeneous solutions, synchronized oscillations,
bumps, chaotic dynamics, wave or bump splitting are exhibited and arise from
static or dynamic Turing-Hopf bifurcations. These surprising phenomena allow
further exploring the role of noise in the nervous system.Comment: Updated to the latest version published, and clarified the dependence
in space of Brownian motion
Delayed feedback control of self-mobile cavity solitons in a wide-aperture laser with a saturable absorber
We investigate the spatiotemporal dynamics of cavity solitons in a broad area
vertical-cavity surface-emitting laser with saturable absorption subjected to
time-delayed optical feedback. Using a combination of analytical, numerical and
path continuation methods we analyze the bifurcation structure of stationary
and moving cavity solitons and identify two different types of traveling
localized solutions, corresponding to slow and fast motion. We show that the
delay impacts both stationary and moving solutions either causing drifting and
wiggling dynamics of initially stationary cavity solitons or leading to
stabilization of intrinsically moving solutions. Finally, we demonstrate that
the fast cavity solitons can be associated with a lateral mode-locking regime
in a broad-area laser with a single longitudinal mode
Bifurcation analysis of a normal form for excitable media: Are stable dynamical alternans on a ring possible?
We present a bifurcation analysis of a normal form for travelling waves in
one-dimensional excitable media. The normal form which has been recently
proposed on phenomenological grounds is given in form of a differential delay
equation. The normal form exhibits a symmetry preserving Hopf bifurcation which
may coalesce with a saddle-node in a Bogdanov-Takens point, and a symmetry
breaking spatially inhomogeneous pitchfork bifurcation. We study here the Hopf
bifurcation for the propagation of a single pulse in a ring by means of a
center manifold reduction, and for a wave train by means of a multiscale
analysis leading to a real Ginzburg-Landau equation as the corresponding
amplitude equation. Both, the center manifold reduction and the multiscale
analysis show that the Hopf bifurcation is always subcritical independent of
the parameters. This may have links to cardiac alternans which have so far been
believed to be stable oscillations emanating from a supercritical bifurcation.
We discuss the implications for cardiac alternans and revisit the instability
in some excitable media where the oscillations had been believed to be stable.
In particular, we show that our condition for the onset of the Hopf bifurcation
coincides with the well known restitution condition for cardiac alternans.Comment: to be published in Chao
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