Gaussian perturbations of circle maps: A spectral approach

@article{Mayberry2009GaussianPO,
  title={Gaussian perturbations of circle maps: A spectral approach},
  author={J. P. Mayberry},
  journal={Annals of Applied Probability},
  year={2009},
  volume={19},
  pages={1143-1171}
}
  • J. Mayberry
  • Published 1 June 2009
  • Mathematics
  • Annals of Applied Probability
In this work, we examine spectral properties of Markov transition operators corresponding to Gaussian perturbations of discrete time dynamical systems on the circle. We develop a method for calculating asymptotic expressions for eigenvalues (in the zero noise limit) and show that changes to the number or period of stable orbits for the deterministic system correspond to changes in the number of limiting modulus 1 eigenvalues of the transition operator for the perturbed process. We call this… 

Figures from this paper

A Spectral Analysis of the Sequence of Firing Phases in Stochastic Integrate-and-Fire Oscillators

Integrate and fire oscillators are widely used to model the generation of action potentials in neurons. In this paper, we discuss small noise asymptotic results for a class of stochastic integrate

References

SHOWING 1-10 OF 15 REFERENCES

Characterization of Stochastic Bifurcations in a Simple Biological Oscillator

This study of the effect of noise on bifurcations in a simple biological oscillator with a periodically modulated threshold uses the first-passage-time problem of the Ornstein–Uhlenbeck process with

Bifurcations of One-Dimensional Stochastic Differential Equations

We consider families of random dynamical systems induced by parametrized one-dimensional stochastic differential equations. We give necessary and sufficient conditions on the invariant measures of

Noise-induced effects on period-doubling bifurcation for integrate-and-fire oscillators.

  • T. Tateno
  • Mathematics
    Physical review. E, Statistical, nonlinear, and soft matter physics
  • 2002
TLDR
The results of the ways in which noise influences period-doubling bifurcation in the parameter space of an integrate-and-fire model with a periodically modulated reset level are illustrated.

A Spectral Analysis of the Sequence of Firing Phases in Stochastic Integrate-and-Fire Oscillators

Integrate and fire oscillators are widely used to model the generation of action potentials in neurons. In this paper, we discuss small noise asymptotic results for a class of stochastic integrate

Stochastic Resonance-Like Behavior in the Sine-Circle Map

In the sine-circle map, the phase-locking structure on the parameter plane (Ω, K )i s known as Arnold’s tongue. The overlapping of two adjacent tongues implies the coexistence of two phase-locked

Probability: Theory and Examples

This book is an introduction to probability theory covering laws of large numbers, central limit theorems, random walks, martingales, Markov chains, ergodic theorems, and Brownian motion. It is a

A problem concerning orthogonal polynomials

In his paper Note on the orthogonality of Tchebycheff polynomials on confocal ellipses,] Walsh has obtained a new orthogonality property of the Tchebycheff polynomials cos k arc cos z arising by

Spectra and pseudospectra : the behavior of nonnormal matrices and operators

spectra and pseudospectra springerlink. spectra and pseudospectra the behavior of nonnormal. spectra and pseudospectra the behavior of nonnormal. spectra and pseudospectra the behavior of nonnormal.