# Partial control of delay-coordinate maps

## TL;DR

This paper demonstrates how partial control methods can be effectively applied to delay-coordinate maps in nonlinear dynamical systems, even with disturbances larger than the control input, including controlling hyperchaotic systems.

## Contribution

It introduces a novel application of partial control to delay-coordinate maps, enabling control of hyperchaotic systems and reducing variability in observed time series.

## Key findings

- Partial control can prevent undesirable system states.
- Control is effective even with disturbances larger than control inputs.
- Hyperchaotic systems can be partially controlled for the first time.

## Abstract

Delay-coordinate maps have been widely used recently to study nonlinear dynamical systems, where there is only access to the time series of one of their variables. Here, we show how the partial control method can be applied in this kind of framework in order to prevent undesirable situations for the system or even to reduce the variability of the observable time series associated with it. The main advantage of this control method, is that it allows to control delay-coordinate maps even if the control applied is smaller than the external disturbances present in the system. To illustrate how it works, we have applied it to three well-known models in Nonlinear Dynamics with different delays such as the two-dimensional cubic map, the standard map and the three-dimensional hyperchaotic H\'enon map. For the first time we show here how hyperchaotic systems can be partially controlled.