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www.nature.com/scientificreports OPEN Controlling nonlinear dynamical systems into arbitrary states using machine learning Alexander Haluszczynski1,2* & Christoph Räth3 Controlling nonlinear dynamical systems is a central task in many diferent areas of science and engineering. Chaotic systems can be stabilized (or chaotifed) 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 diferent chaotic behavior. Having this highly fexible control scheme with little demands on the amount of required data on hand, we briefy discuss possible applications ranging from engineering to medicine. Te possibility to control nonlinear chaotic systems into stable states has been a remarkable discovery1,2. Based on the knowledge of the underlying equations, one can force the system from a chaotic state into a fxed point or periodic orbit by applying an external force. Tis can be achieved based on the pioneering approaches by Ott et al.1 or Pyragas3. In the former, a parameter of the system is slightly changed when it is close to an unstable periodic orbit in phase space, while the latter continuously applies a force based on time delayed feedback. Tere have been many extensions of those basic approaches (see e.g. Boccaletti et al.4 and references therein) including “anti-control” schemes5, that break up periodic or synchronized motion. However, all of them do not allow to control the system into well-specifed, yet more complex target states such as chaotic or intermittent behavior. Further, these methods either require exact knowledge about the system, i.e. the underlying equations of motion, or rely on phase space techniques for which very long time series are necessary. In recent years, tremendous progress has been made in the prediction of nonlinear dynamical systems by means of machine learning (ML). It has been demonstrated that not only exact short-term predictions over several Lyapunov times become possible, but also the long-term behavior of the system (its “climate”) can be reproduced with unexpected accuracy6–12—even for very high-dimensional systems13–15. While several ML tech- niques have successfully been applied to time series prediction, reservoir computing (RC)16,17 can be considered as the so far best approach, as it combines ofen superior performance with intrinsic advantages like smaller network size, higher robustness, fast and comparably transparent learning18 and the prospect of highly efcient hardware realizations19–21. Combining now ML-based predictions of nonlinear systems with manipulation steps, we propose in this study a novel, fully data-driven approach for controlling nonlinear dynamical systems. In contrast to previous methods, this allows to obtain a variety of target states including periodic, intermittent and chaotic ones. Furthermore, we do not require the knowledge of the underlying equations. Instead, it is sufcient to record some history of the system that allows the machine learning method to be sufciently trained. As previously outlined22, an adequate learning requires orders of magnitude less data than phase space methods. 1Department of Physics, Ludwig-Maximilians-Universität, Schellingstraße 4, 80799 Munich, Germany. 2Allianz Global Investors, risklab, Seidlstraße 24, 80335 Munich, Germany. 3Institut für Materialphysik im Weltraum, Deutsches Zentrum für Luft- und Raumfahrt, Münchner Str. 20, 82234 Wessling, Germany. *email: [email protected] Scientifc Reports | (2021) 11:12991 | https://doi.org/10.1038/s41598-021-92244-6 1 Vol.:(0123456789) www.nature.com/scientificreports/ Figure 1. Periodic to chaotic control. Top: 2D attractor representation in the x–y plane. Bottom: X coordinate time series. Lef plots show the original chaotic state which changes to a periodic state (middle) afer tuning the order parameter. Afer applying the control mechanism, the system is forced into a chaotic state again (right). Results We defne 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. Ten, the system behavior changes into another state Y as a consequence of order parameter changes or some uncontrollable external force. Te aim of a control mechanism is now to push the system back into its original state X , while the cause for the initial change in state is still present. Tis can be achieved by deriving a suitable control force F(t) which is applied while the system is in state Y . Deriving F(t) requires the knowledge of how the trajectory u(t) of the system would have evolved if the system was still in state X instead. Tis ’what if’ scenario can be obtained by training a suitable machine learning technique on past observations of the system while being in state X . In this study, this is achieved by using reservoir computing23, which is a recurrent neural network based approach. In principle, any other prediction method could be used instead as long as it is able to deliver good predictions. Once trained and synchronized, it can create predictions v(t) of arbitrary length from which the control force F(t) is derived as F(t) = K(u(t) − v(t)), (1) where K scales the magnitude of the force. Since F(t) only depends on the (measured) coordinates u(t) and the ML prediction v(t), no mathematical model is required to control the system and thus the method is generally applicable as long as good predictions are available. Te defnition of the control force being dependent on the distance between the actual coordinate and a target coordinate is similar to what has been originally proposed by Pyragas3. However, in our case the control is not limited to periodic orbits but can achieve a variety of dynamical target states. A step by step description of the method is given in Section 0.2. Te control of nonlinear dynamical system is studied on the example of the Lorenz system24, which is a model for atmospheric convection. Depending on the choice of parameters, the system exhibits e.g. periodic, intermittent or chaotic behavior. Te equations read x˙ = σ(y − x);˙y = x(ρ − z) − y;˙z = xy − βz, (2) and π ≡ (σ, ρ, β) are the order parameters that lead to a certain state and the trajectory is thus described by T u(t) = (x(t), y(t), z(t)) . First, we simulate the Lorenz system with parameters π such that we obtain the desired initial state X . Second, we train reservoir computing on the resulting trajectory until time step ttrain . Ten, the ∗ parameters are shifed to π such that the system behavior changes to state Y at time step tshift . If tshift ≥ ttrain , the RC system is synchronized accordingly with the trajectory since ttrain . Synchronization means that the scalar states of the reservoir (see Eq. 5) are updated but the system is not re-trained. To control the system now back into state X , the correction force F(t) is derived in each time step based on the prediction v(t) and applied to the system by solving the diferential equations of the system for the next time step including F(t) t+�t ∗ u(t + �t) = (f˙(u(t˜), π ) + F(t˜))dt˜, (3) t where f˙ is defned in Eq. (2). Te knowledge of f˙ is only required for the model system examples in this study but not for real world applications. Te equations are solved using the 4th order Runge–Kutta method with a time ∗ resolution t = 0.02 . Since still the parameters π are used, the system would continue to exhibit the undesired state Y if the control force was 0. For the Lorenz system, the scaling constant set to K = 25. We did not optimize for K and empirically found that our method works for a wide range of choices. It is important to emphasize that a smaller choice for K does not necessarily mean that a smaller force is needed, because smaller values may allow for more separation of u(t) and v(t). Figure 1 shows the results for the Lorenz system originally (left side) being in a chaotic state X ( π =[σ = 10.0, ρ = 167.2, β = 8/3] ), which then changes to periodic behavior (middle) Y afer ρ is changed to ρ = 166 . Ten, the control mechanism is activated and the resulting attractor again resembles the original chaotic state (lef). While ‘chaotifcation’ of periodic states has been achieved in the past, the resulting attrac- tor generally did not correspond to a certain specifed target state but just exhibited some chaotic behavior. Since we would like to not only rely on a visual assessment, we characterize the attractors using quantitative Scientifc Reports | (2021) 11:12991 | https://doi.org/10.1038/s41598-021-92244-6 2 Vol:.(1234567890) www.nature.com/scientificreports/ Largest Lyapunov exponent Correlation dimension ν orig changed controlled νorig νchanged νcontrolled Periodic → Chaotic 0.851 ± 0.070 0.080 ± 0.075 0.841 ± 0.074 1.700 ± 0.065 1.052 ± 0.071 1.700 ± 0.061 Chaotic → Intermittent 0.571 ± 0.096 0.853 ± 0.053 0.614 ± 0.101 1.321 ± 0.086 1.678 ± 0.055 1.351 ± 0.091 ChaoticB → ChaoticA 0.479 ± 0.060 0.643 ± 0.075 0.478 ± 0.067 1.941 ± 0.038 1.948 ± 0.047 1.933 ± 0.040 ChaoticD → ChaoticC 0.819 ± 0.092 0.884 ± 0.058 0.822 ± 0.052 1.855 ± 0.069 1.959 ± 0.037 1.866 ± 0.050 Periodic ← Chaotic - 0.003 ± 0.012 0.844 ± 0.059 0.028 ± 0.110 1.001 ± 0.065 1.700 ± 0.071 1.001 ± 0.061 Chaotic ← Intermittent 0.851 ± 0.070 0.550 ± 0.094 0.828 ± 0.067 1.700 ± 0.086 1.326 ± 0.055 1.698 ± 0.091 ChaoticB ← ChaoticA 0.629 ± 0.069 0.446 ± 0.068 0.629 ± 0.066 1.948 ± 0.037 1.939 ± 0.049 1.956 ± 0.037 ChaoticD ← ChaoticC 0.881 ± 0.092 0.836 ± 0.058 0.880 ± 0.052 1.958 ± 0.069 1.864 ± 0.038 1.951 ± 0.050 Table 1.

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