Controlling transient chaos in the Lorenz system with machine learning
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2025-03-28
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Springer
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This paper presents a novel approach to sustain transient chaos in the Lorenz system through the estimation of safety functions using a transformer-based model. Unlike classical methods that rely on iterative computations, the proposed model directly predicts safety functions without requiring finetuning or extensive system knowledge. The results demonstrate that this approach effectively maintains chaotic trajectories within the desired phase space region, even in the presence of noise, making it a viable alternative to traditional methods. A detailed comparison of safety functions, safe sets, and their control performance highlights the strengths and trade-offs of the two approaches.
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This article presents a novel approach for controlling transient chaos in the Lorenz system using a transformer-based machine learning model. Transient chaos refers to chaotic dynamics that occur temporarily before a system stabilizes into a non-chaotic state. The challenge lies in sustaining this chaotic regime, which is often desirable in applications like energy harvesting or population modeling. Traditional control methods rely on the computation of safety functions to determine minimal control inputs that keep the system within a chaotic regime. However, these methods are computationally intensive and require detailed knowledge of the system’s dynamics.
The proposed method addresses these limitations by leveraging a transformer neural network trained to estimate safety functions directly from noisy trajectory data. This data-driven model eliminates the need for iterative computation and physical system modeling, offering a significant reduction in computational complexity. The study focuses on a reduced one-dimensional map of the Lorenz system, constructed from a Poincaré section, to efficiently apply control and compare results between classical and machine learning approaches.
Performance is evaluated under different noise levels. In both low and high noise scenarios, the machine learning-based safety function successfully confines trajectories within the desired region, maintaining the system in the transient chaotic state. Comparisons show that while the machine learning approach tends to slightly overestimate or underestimate safe regions, it remains effective in practice, requiring only marginally higher control efforts. Visual analysis of safe sets and control trajectories confirms the validity of the method.
The results demonstrate that transformer-based models offer a powerful alternative for chaos control in nonlinear dynamical systems. This technique is particularly useful for systems where traditional modeling is impractical or where rapid, real-time control is necessary. Future improvements could enhance the accuracy of the predicted safe sets, especially in scenarios with complex or highly variable noise. Overall, this work paves the way for scalable, efficient, and generalizable control strategies using modern machine learning tools.
Open Access funding provided thanks to the CRUE-CSIC agreement with Springer Nature.
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Valle, D., Capeans, R., Wagemakers, A. et al. Controlling transient chaos in the Lorenz system with machine learning. Eur. Phys. J. Spec. Top. (2025). https://doi.org/10.1140/epjs/s11734-025-01589-w
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