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

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.

References in corpus (33)

• Lévy walks
• Persistence and First-Passage Properties in Non-equilibrium Systems
• Statistical Aging and Non Ergodicity in the Fluorescence of Single Nanocrystals
• Stochastic Ergodicity Breaking: a Random Walk Approach
• Leaking Chaotic Systems
• Exact Solution for the Influence of Spectral Diffusion on Single-Molecule Photon-Statistics
• Aging Continuous Time Random Walks
• Universal First-passage Properties of Discrete-time Random Walks and Levy Flights on a Line: Statistics of the Global Maximum and Records
• On universality of algebraic decays in Hamiltonian systems
• Pesin-Type Identity for Weak Chaos
• First-passage and first-hitting times of Levy flights and Levy walks
• Open circular billiards and the Riemann hypothesis
• Universality of algebraic laws in Hamiltonian systems
• Ergodicity Breaking in a Deterministic Dynamical System
• Aging generates regular motions in weakly chaotic systems
• Fractal Properties of Anomalous Diffusion in Intermittent Maps
• Subexponential instability implies infinite invariant measure
• Separation of trajectories and its Relation to Entropy for Intermittent Systems with a Zero Lyapunov exponent
• Spectral degeneracy and escape dynamics for intermittent maps with a hole
• Cycle expansions for intermittent maps
• Understanding Anomalous Transport in Intermittent Maps: From Continuous Time Random Walks to Fractals
• Cycle expansions for intermittent diffusion
• Dependence of chaotic diffusion on the size and position of holes
• Escape from intermittent repellers- Periodic orbit theory for crossover from exponential to algebraic decay
• Pesin-type relation for subexponential instability
• Lyapunov statistics and mixing rates for intermittent systems
• Computing the diffusion coefficient for intermittent maps: Resummation of stability ordered cycle expansions
• Numerical estimate of infinite invariant densities: application to Pesin-type identity
• Number of first-passage times as a measurement of information for weakly chaotic systems
• Slow and fast escape for open intermittent maps
• Singularity of Levy walks in the lifted Pomeau-Manneville map
• Escape rate scaling in infinite measure preserving systems
• Escape rates for the Farey map with approximated holes