Avoiding extremes using Partial Control

TL;DR

This paper demonstrates how weak control strategies can prevent chaotic economic systems from reaching extreme states by maintaining the system within safe regions, despite external disturbances.

Contribution

It introduces the concept of safe sets for partial control of chaotic systems and analyzes their behavior under parameter changes.

Findings

Safe sets enable partial control of chaotic dynamics.

Safe sets can vary discontinuously with parameters.

Weak control can prevent extreme fluctuations in chaotic systems.

Abstract

Dynamical systems can be prone to severe fluctuations due to the presence of chaotic dynamics. This paper explains for a toy chaotic economic model how such a system can be regulated by the application of relatively weak control to keep the system confined to a bounded region of the phase space, even in the presence of strong external disturbances. Since the control here is weaker than the disturbance, the system cannot be controlled to a particular trajectory, but under certain circumstances it can be partially controlled to avoid extreme values. Partial control depends on the existence of a certain set called a "safe sets". We describe the safe set and how it varies with parameters, sometimes continuously and sometimes discontinuously.