6,971 research outputs found
Mean Square Polynomial Stability of Numerical Solutions to a Class of Stochastic Differential Equations
The exponential stability of numerical methods to stochastic differential
equations (SDEs) has been widely studied. In contrast, there are relatively few
works on polynomial stability of numerical methods. In this letter, we address
the question of reproducing the polynomial decay of a class of SDEs using the
Euler--Maruyama method and the backward Euler--Maruyama method. The key
technical contribution is based on various estimates involving the gamma
function
Phase diffusion as a model for coherent suppression of tunneling in the presence of noise
We study the stabilization of coherent suppression of tunneling in a driven
double-well system subject to random periodic function ``kicks''. We
model dissipation due to this stochastic process as a phase diffusion process
for an effective two-level system and derive a corresponding set of Bloch
equations with phase damping terms that agree with the periodically kicked
system at discrete times. We demonstrate that the ability of noise to localize
the system on either side of the double-well potenital arises from overdamping
of the phase of oscillation and not from any cooperative effect between the
noise and the driving field. The model is investigated with a square wave
drive, which has qualitatively similar features to the widely studied
cosinusoidal drive, but has the additional advantage of allowing one to derive
exact analytic expressions.Comment: 17 pages, 4 figures, submitted to Phys. Rev.
Optimizing periodicity and polymodality in noise-induced genetic oscillators
Many cellular functions are based on the rhythmic organization of biological
processes into self-repeating cascades of events. Some of these periodic
processes, such as the cell cycles of several species, exhibit conspicuous
irregularities in the form of period skippings, which lead to polymodal
distributions of cycle lengths. A recently proposed mechanism that accounts for
this quantized behavior is the stabilization of a Hopf-unstable state by
molecular noise. Here we investigate the effect of varying noise in a model
system, namely an excitable activator-repressor genetic circuit, that displays
this noise-induced stabilization effect. Our results show that an optimal noise
level enhances the regularity (coherence) of the cycles, in a form of coherence
resonance. Similar noise levels also optimize the multimodal nature of the
cycle lengths. Together, these results illustrate how molecular noise within a
minimal gene regulatory motif confers robust generation of polymodal patterns
of periodicity.Comment: 9 pages, 6 figure
Stabilisation and destabilisation of nonlinear differential equations by noise
This paper considers the stabilisation and destabilisa- tion by a Brownian noise perturbation which preserves the equilibrium of the ordinary dierential equation x0(t) = f(x(t)). In an extension of earlier work, we lift the restriction that f obeys a global linear bound, and show that when f is locally Lipschitz, a function g can always be found so that the noise perturbation g(X(t)) dB(t) either stabilises an unstable equilibrium, or destabilises a stable equilibrium. When the equilibrium of the deterministic equation is non{hyperbolic, we show that a non{hyperbolic perturbation suffices to change the stability properties of the solution.
- …