Gaussian perturbations of circle maps: A spectral approach

TL;DR

This paper analyzes how Gaussian noise affects the spectral properties of circle maps, revealing a phenomenon called λ-bifurcation where stable orbit changes correspond to eigenvalue shifts.

Contribution

It introduces a spectral method to compute eigenvalues and eigenfunctions of perturbed circle maps, linking bifurcations to eigenvalue changes in the transition operator.

Findings

Eigenvalues asymptotically calculated in zero noise limit

Stable orbit changes correspond to eigenvalue shifts

Eigenfunctions related to Hermite functions

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 $\lambda$-bifurcation. Asymptotic expressions for the corresponding eigenfunctions and eigenmeasures are also derived and are related to Hermite functions.