Partially controlling transient chaos in the Lorenz equations

TL;DR

This paper demonstrates how partial control techniques can prevent escape from transient chaos in the Lorenz system, using 1D, 2D, and 3D map-based implementations to keep trajectories within chaotic regions.

Contribution

It introduces and compares three different methods for applying partial control to the Lorenz system to maintain chaotic behavior indefinitely.

Findings

Partial control successfully prevents escape in the Lorenz system.

Three implementation methods are effective: 1D maxima map, 2D Poincaré map, 3D fixed-interval map.

The 3D map offers practical advantages for real-world applications.

Abstract

Transient chaos is a characteristic behavior in nonlinear dynamics where trajectories in a certain region of phase space behave chaotically for a while, before escaping to an external attractor. In some situations the escapes are highly undesirable, so that it would be necessary to avoid such a situation. In this paper we apply a control method known as partial control that allows one to prevent the escapes of the trajectories to the external attractors, keeping the trajectories in the chaotic region forever. To illustrate how the method works, we have chosen the Lorenz system for a choice of parameters where transient chaos appears, as a paradigmatic example in nonlinear dynamics. We analyze three quite different ways to implement the method. First, we apply this method by building a 1D map using the successive maxima of one of the variables. Next, we implement it by building a 2D map through a Poincar\'{e} section. Finally, we built a 3D map, which has the advantage of using a fixed time interval between application of the control, which can be useful for practical applications.