Abstract
Controlling nonlinear dynamical systems is a central task in many different areas of science and engineering. Chaotic systems can be stabilized (or chaotified) with small perturbations, yet existing approaches either require knowledge about the underlying system equations or large data sets as they rely on phase space methods. In this work we propose a novel and fully data driven scheme relying on machine learning (ML), which generalizes control techniques of chaotic systems without requiring a mathematical model for its dynamics. Exploiting recently developed ML-based prediction capabilities, we demonstrate that nonlinear systems can be forced to stay in arbitrary dynamical target states coming from any initial state. We outline and validate our approach using the examples of the Lorenz and the Rössler system and show how these systems can very accurately be brought not only to periodic, but even to intermittent and different chaotic behavior. Having this highly flexible control scheme with little demands on the amount of required data on hand, we briefly discuss possible applications ranging from engineering to medicine.
Highlights
The largest Lyapunov exponent of the original chaotic system orig = 0.851 significantly reduces to changed = 0.080 when the parameter change drives the system into a periodic state
In nonlinear technical systems such as rocket engines it can be used to prevent the engine from critical combustion instabilities[27,28]. This could be achieved by detecting them based on the reservoir computing predictions and subsequently controlling the system into a more stable state
The heart of a healthy human does not beat in a purely periodic fashion but rather shows features being typical for chaotic systems like m ultifractality[29] that vary significantly among individuals
Summary
We define the situation that requires to be controlled in the following way: A dynamical system with trajectory u is in state X , which may represent e.g. periodic, intermittent or chaotic behavior. The control mechanism is activated and the resulting attractor again resembles the original chaotic state (left).
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