10,081 research outputs found
Exit time asymptotics for small noise stochastic delay differential equations
Dynamical system models with delayed dynamics and small noise arise in a
variety of applications in science and engineering. In many applications,
stable equilibrium or periodic behavior is critical to a well functioning
system. Sufficient conditions for the stability of equilibrium points or
periodic orbits of certain deterministic dynamical systems with delayed
dynamics are known and it is of interest to understand the sample path behavior
of such systems under the addition of small noise. We consider a small noise
stochastic delay differential equation (SDDE) with coefficients that depend on
the history of the process over a finite delay interval. We obtain asymptotic
estimates, as the noise vanishes, on the time it takes a solution of the
stochastic equation to exit a bounded domain that is attracted to a stable
equilibrium point or periodic orbit of the corresponding deterministic
equation. To obtain these asymptotics, we prove a sample path large deviation
principle (LDP) for the SDDE that is uniform over initial conditions in bounded
sets. The proof of the uniform sample path LDP uses a variational
representation for exponential functionals of strong solutions of the SDDE. We
anticipate that the overall approach may be useful in proving uniform sample
path LDPs for a broad class of infinite-dimensional small noise stochastic
equations.Comment: 39 page
Relaxation oscillations, pulses, and travelling waves in the diffusive Volterra delay-differential equation
The diffusive Volterra equation with discrete or continuous delay is studied in the limit of long delays using matched asymptotic expansions. In the case of continuous delay, the procedure was explicitly carried out for general normalized kernels of the form Sigma/sub n=p//sup N/ g/sub n/(t/sup n//T/sup n+1/)e/sup -t/T/, pges2, in the limit in which the strength of the delayed regulation is much greater than that of the instantaneous one, and also for g/sub n/=delta/sub n2/ and any strength ratio. Solutions include homogeneous relaxation oscillations and travelling waves such as pulses, periodic wavetrains, pacemakers and leading centers, so that the diffusive Volterra equation presents the main features of excitable media
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
- …