The Annals of Applied Probability

Gaussian perturbations of circle maps: A spectral approach

John Mayberry

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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

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

First available in Project Euclid: 15 June 2009

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Zentralblatt MATH identifier

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]

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


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

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