Follow the fugitive: an application of the method of images to open dynamical systems

Abstract

Borrowing and extending the method of images we introduce a theoretical framework that greatly simplifies analytical and numerical investigations of the escape rate in open dynamical systems. As an example, we explicitly derive the exact size- and position-dependent escape rate in a Markov case for holes of finite size. Moreover, a general relation between the transfer operators of closed and corresponding open systems, together with the generating function of the probability of return to the hole is derived. This relation is then used to compute the small hole asymptotic behavior, in terms of readily calculable quantities. As an example we derive logarithmic corrections in the second order term. Being valid for Markov systems, our framework can find application in information theory, network theory, quantum Weyl law and via Ulam's method can be used as an approximation method in more general dynamical systems.

9 pages, 1 figure

References in corpus (5)

• Leaking Chaotic Systems
• Noise-enhanced trapping in chaotic scattering
• Peeping at chaos: Nondestructive monitoring of chaotic systems by measuring long-time escape rates
• Faster than expected escape for a class of fully chaotic maps
• Fractal diffusion coefficient from dynamical zeta functions

Cited by in corpus (6)

• Chaotic Systems with Absorption
• Escape through a time-dependent hole in the doubling map
• Escape rate scaling in infinite measure preserving systems
• Temporal-varying failures of nodes in networks
• Escape rates for special flows and their higher order asymptotics
• Escape rates for the Farey map with approximated holes