Controlling Escape in the Standard Map

TL;DR

This paper explores how controlling small regions in the phase space of the Standard Map can effectively reduce the diffusion exponent, especially near bifurcation points, by targeting hyperbolic points and escape channels.

Contribution

It demonstrates that controlling hyperbolic points and escape channels in the Standard Map significantly decreases the diffusion exponent, providing a targeted approach to phase space control.

Findings

Controlling hyperbolic points reduces the diffusion exponent more effectively.

Larger controlled areas lead to smaller diffusion exponents.

Targeted control near bifurcation points outperforms random control.

Abstract

We investigate how the diffusion exponent is affected by controlling small domains in the phase space.The main Kolomogorov-Arnold-Moser - KAM island of the Standard Map is considered to validate the investigation. The bifurcation scenario where the periodic island emits smaller resonance regions is considered and we show how closing paths escape from the island shore by controlling points and hence making the diffusion exponent smaller. We notice the bigger controlled area the smaller the diffusion exponent. We show that controlling around the hyperbolic points associated to the bifurcation is better than a random control to reduce the diffusion exponent. The recurrence plot shows us channels of escape and a control applied there reduces the diffusion exponent.