arXiv subjects

Peter H. Baxendale

Publications and source records attributed to Peter H. Baxendale.

4 recordsLinked to original sources

arXiv2023

Lyapunov exponents and shear-induced chaos for a Hopf bifurcation with additive noise

This paper considers the effect of additive white noise on the normal form for the supercritical Hopf bifurcation in 2 dimensions. The main results involve the asymptotic behavior of the top Lyapunov exponent lambda associated with this random dynamical system as one or more of the parameters in the system tend to 0 or infinity. This enables the construction of a bifurcation diagram in parameter space showing stable regions where lambda is negative (implying synchronization) and unstable regions where lambda is positive (implying chaotic behavior). The value of lambda depends strongly on the shearing effect of the twist factor of the deterministic Hopf bifurcation. If the twist factor is sufficiently small then lambda is negative regardless of all the other parameters in the system. But when all the parameters except the shear coefficient are fixed then lambda tends to infinity as the shear coefficient tends to infinity.

math.DS↗

arXiv2023

Almost-Sure Stability of the Single Mode Solution of a Noisy Nonlinear Autoparametric System

For a pendulum suspended below a vibrating block with white noise forcing, the solution in which the pendulum remains vertical is called the single mode solution. When this solution becomes unstable there is energy transfer from the block to the pendulum, helping to absorb the vibrations of the block. We study the Lyapunov exponent governing the almost-sure stability of the process linearized along the single mode solution. The linearized equation is excited by a combination of white and colored noise processes, which makes the evaluation of the Lyapunov exponent non trivial. We obtain an explicit third order asymptotic expression for the Lyapunov exponent as the intensity of the white noise forcing tends to zero.

math.PR↗

arXiv2019

Noise Sharing and Mexican Hat Coupling in a Stochastic Neural Field

A diffusion-type coupling operator biologically significant in neuroscience is a difference of Gaussian functions (Mexican Hat operator) used as a spatial-convolution kernel. We are interested in pattern formation by \emph{stochastic} neural field equations, a class of space-time stochastic differential-integral equations using the Mexican Hat kernel. We explore, quantitatively, how the parameters that control the shape of the coupling kernel, coupling strength, and aspects of spatially-smoothed space-time noise, influence the pattern in the resulting evolving random field. We confirm that a spatial pattern that is damped in time in a deterministic system may be sustained and amplified by stochasticity. We find that spatially-smoothed noise alone causes pattern formation even without direct spatial coupling. Our analysis of the interaction between coupling and noise sharing allows us to determine parameter combinations that are optimal for the formation of spatial pattern.

q-bio.NC↗

arXiv2005

Renewal theory and computable convergence rates for geometrically ergodic Markov chains

We give computable bounds on the rate of convergence of the transition probabilities to the stationary distribution for a certain class of geometrically ergodic Markov chains. Our results are different from earlier estimates of Meyn and Tweedie, and from estimates using coupling, although we start from essentially the same assumptions of a drift condition toward a ``small set.'' The estimates show a noticeable improvement on existing results if the Markov chain is reversible with respect to its stationary distribution, and especially so if the chain is also positive. The method of proof uses the first-entrance-last-exit decomposition, together with new quantitative versions of a result of Kendall from discrete renewal theory.

math.PR↗