Samuel Brevitt

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

Chaotic dynamical systems may be characterised by a positive Lyapunov exponent, which measures the 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 (for example stretched exponential) in time; and therefore in such cases the Lyapunov exponent vanishes. When a hole is introduced in chaotic systems, the Lyapunov exponent on the system's fractal repeller can be related to the generation of entropy and the escape rate from the system via the escape rate formalism, but no suitable generalisation exists to weakly chaotic systems.

In this work we show that in a paradigmatic one-dimensional weakly chaotic iterated map, the Pomeau-Manneville map, the generation of Lyapunov 'stretching' (a generalisation of the exponent) is completely suppressed in the presence of a hole. These results are based on numerical evidence and a corresponding stochastic model. Our findings are shown to be in line with known mathematical results concerning the collapse of the system's density as it evolves in time. Finally we conclude that, as a result, no suitable generalisation of the escape rate formalism to weakly chaotic systems can exist.