Deep learning-based analysis of basins of attraction
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2024-03-04
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American Institute of Physics
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Resumen
This research addresses the challenge of characterizing the complexity and unpredictability of basins within various dynamical systems. The main focus is on demonstrating the efficiency of convolutional neural networks (CNNs) in this field. Conventional methods become computationally demanding when analyzing multiple basins of attraction across different parameters of dynamical systems. Our research presents an innovative approach that employs CNN architectures for this purpose, showcasing their superior performance in comparison to conventional methods. We conduct a comparative analysis of various CNN models, highlighting the effectiveness of our proposed characterization method while acknowledging the validity of prior approaches. The findings not only showcase the potential of CNNs but also emphasize their significance in advancing the exploration of diverse behaviors within dynamical systems.
Descripción
This article investigates the application of deep learning techniques to the characterization of basins of attraction in nonlinear dynamical systems. Basins of attraction represent sets of initial conditions in phase space that lead to particular attractors. Their structure often exhibits complex and fractal boundaries, especially in multistable or chaotic systems. The unpredictability inherent in such systems can be quantified using several metrics, including fractal dimension, basin entropy, boundary basin entropy, and the Wada property.
Conventional methods for computing these metrics, although accurate, are computationally demanding due to the complexity of nonlinear operations involved. To address this, the study proposes a data-driven approach using convolutional neural networks (CNNs) to predict these metrics directly from images of basins. Several standard CNN architectures (AlexNet, VGG16, VGG19, GoogLeNet, and ResNet50) were trained and evaluated on datasets derived from five well-known dynamical systems: the Duffing oscillator, Newton fractal, forced damped pendulum, Hénon-Heiles system, and magnetic pendulum.
Each CNN was trained separately for each metric using pre-labeled basins. The ResNet50 architecture emerged as the best performer, offering the lowest prediction error across all metrics while maintaining a fast computation time. It provided a significant speed-up—over 300 times faster—compared to traditional methods, with minimal sacrifice in accuracy. This makes it particularly useful for large-scale analysis or real-time applications where computational efficiency is crucial.
The study highlights the effectiveness of CNNs in replacing conventional methods for the quantitative characterization of chaotic systems. By dramatically reducing computational cost while preserving precision, the approach opens new possibilities for exploring dynamical behaviors across a wide range of systems.
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Citación
David Valle, Alexandre Wagemakers, Miguel A. F. Sanjuán; Deep learning-based analysis of basins of attraction. Chaos 1 March 2024; 34 (3): 033105.