Controlling transient chaos in the Lorenz system with machine learning

Eur. Phys. J. Spec. Top. https://doi.org/10.1140/epjs/s11734-025-01589-w THE EUROPEAN PHYSICAL JOURNAL SPECIAL TOPICS Regular Article Controlling transient chaos in the Lorenz system with machine learning David Vallea , Rubén Capeansb , Alexandre Wagemakersc , and Miguel A. F. Sanjuánd Nonlinear Dynamics, Chaos and Complex Systems Group, Departamento de Fı́sica, Universidad Rey Juan Carlos, Tulipán s/n, Móstoles, Madrid 28933, Spain Received 28 January 2025 / Accepted 14 March 2025 © The Author(s) 2025 Abstract 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. 1 Introduction Chaos theory has long captivated researchers due to its intricate dynamics and wide-ranging implications across fields such as meteorology, engineering, and biology [1]. Among its foundational models, the Lorenz system stands out as a key representation of chaotic behavior [2]. Originally developed to model atmospheric convection, the Lorenz system comprises three coupled nonlinear differential equations, showing sensitivity to initial conditions and the presence of invariant structures like chaotic attractors. Transient chaos is a notable phenomenon in chaos theory, characterized by systems temporarily exhibiting chaotic behavior before settling into a final state. In many cases, transient chaos can be beneficial. For instance, chaotic vibrations can enhance the efficiency of energy harvesters [3]. Similarly, chaotic dynamics in ecological models can stabilize populations and prevent collapse [4]. To address the challenge of managing transient chaos, researchers have developed techniques like partial control, which rely on safe sets and safety functions [4], to confine chaotic trajectories within desired regions of phase space [5, 6]. The safety function is an essential tool for confining chaotic transients in systems described by difference equations. By determining the minimal control needed to keep trajectories within a specified phase-space region over a certain number of iterations [7, 8], it provides a robust method for chaos control. While controlling noise-free systems is relatively simple, achieving low control efforts in noisy scenarios presents a greater challenge. However, computing the safety function is resource-intensive, particularly for high-dimensional systems or real-time applications. This computational complexity limits its broader application in scenarios where rapid solutions are critical. Recent advances in machine learning have introduced efficient alternatives for estimating safety functions [9]. Transformer-based models, in particular, have shown great promise in estimating the convergence behavior of the safety function for a wide range of dynamical systems without the need for fine-tuning. These models significantly improve computational efficiency and bypass the requirement of a physical model by relying solely on samples of trajectories from the dynamical system. a e-mail: e-mail: c e-mail: d e-mail: (corresponding author) b 0123456789().: V,-vol 123 Eur. Phys. J. Spec. Top. In this paper, we explore the application of the transformer-based model proposed in [9] for estimating safety functions and safe sets in the one-dimensional Lorenz system, a simplified yet representative model of the threedimensional Lorenz system. This approach leverages machine learning to provide accurate predictions without requiring extensive system knowledge or iterative computations, offering a novel perspective on controlling chaotic systems. By comparing the machine learning-based safety functions with those obtained from classical methods, we demonstrate the potential of this approach to enhance control strategies and provide new insights into transient chaos management. The paper is organized as follows: Sect. 2 provides an overview of the Lorenz system, detailing its dynamics and the construction of the one-dimensional map used for control. Section 3 outlines the computation of safety functions using both classical and machine learning-based methods. Section 4 presents the results, including a detailed comparison of the safety functions and their derived safe sets, control efforts, and trajectory behaviors for both approaches. Finally, Sect. 5 discusses the key findings, limitations of the study, and potential directions for future research. 2 The Lorenz system The Lorenz system, introduced by Edward Lorenz in 1963 [2], is a canonical example of chaotic dynamics in nonlinear systems. It consists of three coupled nonlinear differential equations: ẋ = σ(y − x), ẏ = x(r − z) − y, ẋ = xy − bz. (1) Here, the dot notation (e.g., ẋ) indicates differentiation with respect to time. Depending on σ, r , and b values, the system can exhibit periodic solutions, chaotic attractors, or transient chaos. For this study, we set σ = 10, b = 8/3, and r = 20, a regime known to produce transient chaos [10, 11]. For this choice of parameters, transient chaos in the Lorenz system is characterized by chaotic trajectories that eventually decay into one of two fixed-point attractors, denoted as C + and C − . Figure 1 illustrates this behavior, where the green lines represent transient chaotic orbits that eventually escape into either the red or blue fixed-point attractors. Our goal is to utilize the safety function to prevent trajectories from settling into attractors, ensuring they remain within the transient chaotic regime. Regarding the integration of the orbits, we performed numerical simulations of the Lorenz system using a fourthorder Runge–Kutta integrator. Initial conditions for 2000 trajectories were randomly sampled within the range x, y, z ∈ [−30, 30], with an integration step size of h = 0.001 over a total duration of t = 50 units of time. To incorporate √ noise effects, we added a stochastic term χh at each integration step. The noise term was modeled as χh = η · h · N (0, 1), where η is the noise intensity parameter, h is the integration time step, and N (0, 1) Fig. 1 Illustration of transient chaos and the convergence to fixed-point attractors in the Lorenz system for σ = 10.0, b = 2.67, and r = 20.0. Panel (a) highlights the transient chaotic regime transitioning to the stable orbit near C − (blue), while panel (b) shows the transition to the stable orbit near C + (red). Green curves indicate the transient chaotic trajectories 123 Eur. Phys. J. Spec. Top. Fig. (...truncated)