Investigating the Effects of Gaussian Perturbations on Circle Maps

We investigate whether the classical Devil's staircase of the circle map can be deformed-rather than replacedby a smooth, localized perturbation.Using Python simulations, we compute rotation numbers from long unwrapped phase iterates while plotting dynamics with the wrapped phase, and sweep the driving frequency over [0,1], and visualize time series, staircases, and a (Ω,K) rotation-number map.We then add a positive Gaussian bump to the per-step increment to test the staircase's structural robustness.The unperturbed map reproduces the textbook picture: a monotone staircase with prominent plateaus at low-order rationals and quasiperiodic rises.In contrast, the perturbed map exhibits a near-uniform global lift of the rotation number, suppressing the visible stair-step texture and, near Ω→1, producing an apparent jump toward W≈2.This outcome reflects the non-zero mean of the bump, which reparameterizes transport rather than inducing local plateau widening or edge shifts.The result is both diagnostic and prescriptive: to study genuine deformation, one must employ wrapped, zero-mean (or driverenormalized) perturbations at small amplitude.