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

TL;DR

This paper investigates how introducing a hole affects the chaotic properties of a weakly chaotic system, showing that key measures like the Lyapunov exponent are suppressed, challenging the extension of escape rate formalism to such systems.

Contribution

It demonstrates through numerical and stochastic modeling that the escape rate formalism does not extend to weakly chaotic systems with holes, specifically in the Pomeau-Manneville map.

Findings

Stretching is suppressed in the presence of a hole.

The fractal repeller structure is altered and collapses.

Entropy generation aligns with the collapse of stretching.

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.