Anomaly Detection in Complex Dynamical Systems: A Systematic Framework Using Embedding Theory and Physics-Inspired Consistency

Anomaly detection in complex dynamical systems is essential for ensuring reliability, safety, and efficiency in industrial and cyber-physical infrastructures. Predictive maintenance helps prevent costly failures, while cybersecurity monitoring has become critical as digitized systems face growing threats. Many of these systems exhibit oscillatory behaviors and bounded motion, requiring anomaly detection methods that capture structured temporal dependencies while adhering to physical consistency principles. In this work, we propose a system-theoretic approach to anomaly detection, grounded in classical embedding theory and physics-inspired consistency principles. We build upon the Fractal Whitney Embedding Prevalence Theorem that extends traditional embedding techniques to complex system dynamics. Additionally, we introduce state-derivative pairs as an embedding strategy to capture system evolution. To enforce temporal coherence, we develop a Temporal Differential Consistency Autoencoder (TDC-AE), incorporating a TDC-Loss that aligns the approximated derivatives of latent variables with their dynamic representations. We evaluate our method on two subsets (FD001, FD003) of the C-MAPSS dataset, a benchmark for turbofan engine degradation. TDC-AE machtes LSTMs and outperforms Transformers while achieving a nearly 100x reduction in MAC operations, making it particularly suited for lightweight edge computing. Our findings support the hypothesis that anomalies disrupt stable system dynamics, providing a robust signal for anomaly detection.