The Annals of Applied Probability

Gaussian perturbations of circle maps: A spectral approach

John Mayberry

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Abstract

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 phenomenon a λ-bifurcation. Asymptotic expressions for the corresponding eigenfunctions and eigenmeasures are also derived and are related to Hermite functions.

Article information

Source
Ann. Appl. Probab., Volume 19, Number 3 (2009), 1143-1171.

Dates
First available in Project Euclid: 15 June 2009

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1245071022

Digital Object Identifier
doi:10.1214/08-AAP573

Mathematical Reviews number (MathSciNet)
MR2537202

Zentralblatt MATH identifier
1189.60130

Subjects
Primary: 60J05: Discrete-time Markov processes on general state spaces 37H20: Bifurcation theory [See also 37Gxx]
Secondary: 47A55: Perturbation theory [See also 47H14, 58J37, 70H09, 81Q15]

Keywords
Random perturbations Markov chains transition operators stochastic bifurcations integrate-and-fire models eigenvalues pseudospectra

Citation

Mayberry, John. Gaussian perturbations of circle maps: A spectral approach. Ann. Appl. Probab. 19 (2009), no. 3, 1143--1171. doi:10.1214/08-AAP573. https://projecteuclid.org/euclid.aoap/1245071022


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References

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