Recurrence and Stickiness in the Noisy Harper Map

TL;DR

This work analyzes how additive noise reshapes transport in the area-preserving Harper map by modifying the Poincaré recurrence statistic (PRS). It demonstrates that noise can unlock interior island dynamics, producing extended tails and enhanced stickiness at intermediate times, while sufficiently strong noise drives the long-time PRS back to exponential decay. A simple three-state Markov model explains the emergence of a new slow decaying mode once islands become accessible due to noise. The findings illuminate stochastic perturbations of mixed regular-chaotic Hamiltonian systems and provide a mechanistic link between island structure, noise level, and recurrence tails, with implications for chaotic transport in higher-dimensional settings.

Abstract

When three types of noise are introduced to the area-preserving Harper map, the Poincaré recurrence statistic (PRS) exhibits an extended tail, corresponding to an increased probability of longer recurrence times. For a deterministic case with a mixture of regular and chaotic orbits, regular islands are responsible for a power-law decay in the recurrence distribution. Noise perturbations allow trajectories to access the interior of the islands, and this can enhance their trapping effect, causing many orbits to take longer to return to a neighborhood of their initial conditions and resulting in a slower power-law decay on an intermediate time scale. On a longer time scale, however, the noisy PRS exhibits exponential decay, eventually falling below the deterministic PRS. We compare distributions of trapping and visit times to islands with recurrence times to show the importance of noise in creating tails in the PRS. A simple model of the dynamics -- a Markov chain with three states -- demonstrates how the slower decay can be caused by noise allowing entry to a previously inaccessible island.

Figures (11)

• Figure 1: Phase portraits for the Harper map (\ref{['eq:HarperMap']}) with (a) $(K,L) = (0.6,4)$, and (b) $(K,L) = (1,5)$. The boxes $\Lambda$, (\ref{['eq:StartBox']}) (yellow), ${\cal B}_{11}$, ${\cal B}_{12}$, (\ref{['eq:IslandBoxes']}) (green), and ${\cal B}_{21}$, ${\cal B}_{22}$ (\ref{['eq:SecondIsland']}) (blue) used in § \ref{['sec:Recurrence']} and § \ref{['sec:Stickiness']} are also shown. For (a), there are $100$ initial conditions on the line $x_0 = y_0$ and each is iterated $t = 3(10)^4$ steps. For (b), the $200$ initial conditions are in the box $\Lambda$ close to the fixed point $(0.5,0)$, on the line segment $0.4 \leq x_0 \leq 0.6$, $y_0=0$ iterated to $t = 3000$. A period-two island chain is identified in red.
• Figure 2: Poincaré recurrence statistic (\ref{['eq:PRS']}) for the Harper map with $(K,L)=(0.6,4)$: (a) Gaussian noise and (b) uniform noise. Here we randomly choose $N(0) = 10^8$ initial conditions in $\Lambda$ (\ref{['eq:StartBox']}). Curves are colored according to the variance $\sigma^2$. Note that the deterministic cases (dashed) in the two panels differ slightly due to a different random seed for the choice of initial conditions in $\Lambda$.
• Figure 3: Comparison of the PRS for (\ref{['eq:HarperMap']}) with $(K,L) = (0.6,4.0)$ and three types of noise: Gaussian (\ref{['eq:Gaussian']}) (short dashes), uniform (\ref{['eq:Uniform']}) (long dashes) and post hoc (\ref{['eq:PostHoc']}) (solid) for three values of the variance. The curves are almost indistinguishable.
• Figure 4: Poincaré recurrence statistic $P(t;\Lambda)$ for the Harper map (\ref{['eq:HarperMap_noise']}) with $(K,L)=(1,5)$ and $\Lambda$ in (\ref{['eq:StartBox']}). Both panels use Gaussian noise with variance $\sigma^2$. For panel (a) the noise is applied as in (\ref{['eq:HarperMap_noise']}) and, for panel (b) as in (\ref{['eq:PostHoc']}).
• Figure 5: Histograms of trapping time, $t^T$ in ${\cal B}_1$ (\ref{['eq:IslandBoxes']}), versus recurrence time, $t^R$ to $\Lambda$ (\ref{['eq:StartBox']}), for (\ref{['eq:HarperMap_noise']}) with $(K,L)=(1,5)$. The four panels correspond to variances (a) $\sigma^2 = 0$ (deterministic case), (b) $10^{-8}$, (c) $4(10)^{-8}$, and (d) $10^{-5}$ with Gaussian noise (\ref{['eq:Gaussian']}). The dashed black line is $t^T = t^R$.