Does an intermittent dynamical system remain (weakly) chaotic after drilling in a hole?

TL;DR

The study investigates whether opening a hole in the weakly chaotic Pomeau-Manneville map allows a generalised escape-rate formalism to relate chaos quantities to transport. By examining the Lyapunov stretching $\Lambda_n$ and entropy on trajectories that survive the hole, and by constructing a stochastic Gaspard-Wang model alongside a partially absorbing variant, the authors show that the total stretching remains bounded and the fractal repeller collapses onto pre-images of the marginal fixed point. This leads to the conclusion that no generalised escape-rate formula exists for open weakly chaotic systems, with implications that cross-linking anomalous transport to weak chaos quantities via escape rates is not feasible in this setting. The results are supported by numerical simulations and analytical arguments, and suggest broader instability of weakly chaotic dynamics to hole perturbations, motivating further study in more realistic systems and experimental contexts.

Abstract

Chaotic dynamical systems are often characterised by a positive Lyapunov exponent, which signifies an exponential rate of separation of nearby trajectories. However, in a wide range of so-called weakly chaotic systems, the separation of nearby trajectories is sub-exponential in time, and the Lyapunov exponent vanishes. When a hole is introduced in chaotic systems, the positive Lyapunov exponents on the system's fractal repeller can be related to the generation of metric entropy and the escape rate from the system. The escape rate, in turn, cross-links these two chaos properties to important statistical-physical quantities like the diffusion coefficient. However, no suitable generalisation of this escape rate formalism exists for weakly chaotic systems. In our paper we show that in a paradigmatic one-dimensional weakly chaotic iterated map, the Pomeau-Manneville map, a generalisation of its Lyapunov exponent (which we call `stretching') is completely suppressed in the presence of a hole. This result is based on numerical evidence and a corresponding stochastic model. The correspondence between map and model is tested via a related partially absorbing map. We examine the structure of the map's fractal repeller, which we reconstruct via a simple algorithm. Our findings are in line with rigorous mathematical results concerning the collapse of the system's density as it evolves in time. We also examine the generation of entropy in the open map, which is shown to be consistent with the collapsed stretching. As a result, we conclude that no suitable generalisation of the escape rate formalism to weakly chaotic systems can exist.

Figures (7)

• Figure 1: Left: A typical trajectory of the closed Pomeau-Manneville map as a function of $n$, displaying the periods of chaotic and laminar motions which give it the descriptor 'intermittent'. Right: The closed PM map (red), with a sample trajectory shown as a cobweb plot (green). The line $x_{n+1}=x_n$ is shown for comparison (black).
• Figure 2: Left: The closed Pomeau-Manneville map shown in red for $z=3$, $a=1$, with its corresponding Gaspard-Wang partition shown in blue. The diagonal $x_{n+1}=x_n$ is shown in black. Right: The open Pomeau-Manneville map for $z=3$, $c_1=\frac{2}{3}$, $c_2=\frac{3}{4}$, with its corresponding adapted Gaspard-Wang partition, with the escape region $(c_1,c_2)$ shown in grey.
• Figure 3: Left: In shades of blue (solid), the cumulated Lyapunov stretching $\langle\Lambda_n\rangle_t$ computed from numerical simulations, as a function of $n$, for values of $t=10^i$, $i\in\{2,3,4,5,6\}$, $0\leq n\leq t$. Simulations are averaged over an ensemble of $N=10^4$ surviving trajectories. In shades of red (dashed), analytic estimates to $\langle \Lambda_n \rangle$ derived from the Gaspard-Wang theory, see legend and text for details. Right: In shades of blue (top), the cumulated Lyapunov stretching $\langle\Lambda_n\rangle_t$ incurred during complete (dashed) and incomplete (dotted, bottom) phases of each trajectory, with the total shown in solid lines. In red (below), estimates of the same, derived analytically from Gaspard-Wang theory, see text for details.
• Figure 4: A schematic diagram showing the separation of a typical trajectory into complete and incomplete segments. In blue: the instantaneous rate of stretching $\log M'(x_n)$. By estimating the 'overhang' period $m$, we may estimate the contribution incurred during the incomplete phase.
• Figure 5: In shades of blue: the LZ77 complexity $K(n,t)$ for $t=10^i$, $i\in\{2,3,4,5\}$, averaged over an ensemble of $N=10^3$ surviving trajectories. Error bars at the end of each line show the standard error about the sample mean. In faded red, dashed: the line proportional to $\ln(n)$ for comparison.