Attention-Based Multitask Learning with Phase-Space Reconstruction for Early Detection of Instability and Chaos in the Lorenz System

This paper presents an attention-based multitask deep-learning framework for early prediction of instability and chaos in nonlinear dynamical systems. The proposed method integrates phase-space reconstruction (PSR), multitask learning, and attention-enhanced temporal representation learning to provide early warning of transitions from stable behavior to transitional oscillatory behavior and chaos. The Lorenz system is used as the primary benchmark because it exhibits well-defined stable, transitional, and chaotic regimes under parameter variation. Trajectory data are generated through parameter sweeps near the instability region identified using equilibrium eigenvalue analysis. Delay-coordinate embedding is then applied to reconstruct the underlying phase-space dynamics from scalar observations. The reconstructed trajectories are processed using a multitask neural architecture consisting of a forecasting branch and an attention-enhanced temporal classification branch. The forecasting branch predicts short-horizon future dynamics, while the classification branch combines the encoder state with an attention-derived context vector to identify the current dynamical regime. The proposed framework predicts three regimes: stable, transitional oscillatory, and chaotic. The framework is evaluated using grouped cross-validation, class-weighted optimization, independent parameter-region testing, baseline model comparison, multitask gradient-alignment analysis, noise robustness experiments, and cross-system validation. Experimental results on the Lorenz benchmark demonstrate strong predictive capability, achieving a macro F1-score above 0.91 and a macro ROC-AUC of approximately 0.99 while maintaining low forecasting error. Additional experiments show positive gradient alignment between forecasting and classification objectives, indicating cooperative multitask optimization. Noise robustness analysis demonstrates stable performance under noisy measurements, while cross-system validation on the Rossler and Mackey–Glass systems confirms that the learned dynamical representations generalize beyond a single attractor. Computational analysis further shows that the proposed framework maintains low inference latency suitable for near real-time instability monitoring applications. These results demonstrate that the proposed PSR-attention multitask framework provides a robust and computationally efficient early-warning approach for nonlinear systems approaching instability and chaos.