# Stochastic control stabilizing unstable or chaotic maps

## TL;DR

This paper introduces a stochastic control method that stabilizes unstable and chaotic maps by noise, relaxing previous assumptions and broadening the range of applicable maps and parameters, with simulations demonstrating effectiveness.

## Contribution

It presents a new stochastic control approach that stabilizes chaotic maps with fewer restrictions and wider parameter ranges than previous methods.

## Key findings

- Noise can stabilize unstable equilibria.
- Stabilization effectiveness depends on map instability and initial conditions.
- Bounded noise achieves stabilization without initial value limitations.

## Abstract

The paper considers a stabilizing stochastic control which can be applied to a variety of unstable and even chaotic maps. Compared to previous methods introducing control by noise, we relax assumptions on the class of maps, as well as consider a wider range of parameters for the same maps. This approach allows to stabilize unstable and chaotic maps by noise. The interplay between the map properties and the allowed neighbourhood where a solution can start to be stabilized is explored: as instability of the original map increases, the interval of allowed initial conditions narrows. A directed stochastic control aiming at getting to the target neighbourhood almost sure is combined with a controlling noise. Simulations illustrate that for a variety of problems, an appropriate bounded noise can stabilize an unstable positive equilibrium, without a limitation on the initial value.