A methodology that combines Topological Data Analysis (TDA), specifically 0-D sublevel persistence, with Machine Learning (ML) classifiers to distinguish periodic and chaotic regimes directly from time series is proposed, providing an effective alternative for analysing sparse or incomplete time series.
Abstract
Complex dynamical systems often undergo transitions from periodic to chaotic behaviour as bifurcation parameters vary, making timely detection of these changes essential. Conventional approaches based on the maximal Lyapunov exponent (MLE) generally require either knowledge of the governing equations or sufficiently long, uniformly sampled time series. Their performance degrades when the available data are scanty or contain missing observations, making reliable phase-space reconstruction difficult. We propose a methodology that combines Topological Data Analysis (TDA), specifically 0-D sublevel persistence, with Machine Learning (ML) classifiers to distinguish periodic and chaotic regimes directly from time series. Sublevel persistence extracts topological features by analysing the evolution of minima and maxima, revealing repeating signatures for periodic dynamics and more scattered patterns for chaotic dynamics. These features are used to train logistic regression, support vector machine, and k-nearest neighbour classifiers. Hyperparameters are validated using K-fold cross-validation, yielding average classification accuracies exceeding 90%. The trained classifiers provide binary predictions, identifying periodic and chaotic behaviour in previously unseen data. The proposed methodology is evaluated on the Duffing, R\"ossler, and Lorenz systems, where the detected transitions closely agree with those identified using the MLE, demonstrating the reliability of the approach. It is further applied to real-world ECG signals to classify normal and abnormal heartbeats, producing encouraging performance across standard statistical metrics. The proposed framework provides an effective alternative for analysing sparse or incomplete time series and is particularly useful in experimental settings where conventional nonlinear time-series methods are limited.
It is demonstrated that the squared Pearson correlation coefficient provides a simple quantitative criterion for distinguishing chaos from noise directly from observed time-series data.
Reliable anomaly detection in time series is crucial yet challenging owing to the chaotic nature of signals from sources such as financial markets or natural systems. However, most state-of-the-art approaches rely heavily on extensive parameter tuning and large labeled datasets, which are rarely available in production, making them unsuitable for highly dynamic environments. Because field data are predominantly unlabeled, fully unsupervised anomaly detection has become an operational necessity for practical industrial deployment. We present an enhanced, fully autonomous implementation of the Brain-Inspired Cortical Coding (BIC) framework that effectively eliminates the requirement for manual parameter tuning and labeling. The BIC framework combines adaptive clustering with entropy-oriented cortical coding. Unlike most advanced methods, BIC learns directly from unlabeled data and provides stable performance under varying operational conditions. The evaluation was conducted using synthetic Lorenz attractor signals and real-world artificial intelligence for IT operations (AIOps) key performance indicator (KPI) data characterized by rapidly evolving patterns. The comparative experiments included long short-term memory (LSTM), extreme learning machines (ELM), variational autoencoders (VAE), generative adversarial network (GAN)-based models, Isolation Forest, and statistical baselines. Results show BIC outperforms state-of-the-art models, achieving F1-scores of 0.84 and 0.75 under low- and high-noise Lorenz conditions, and an F1-score of approximately 0.80 on the AIOps KPI dataset across both short and long training regimes. Furthermore, under extended evaluation, BIC maintains stable detection performance, whereas competing models show significant deterioration or require additional retraining. These results demonstrate that BIC provides a robust and scalable solution for anomaly detection in noisy, chaotic, and rapidly evolving time-series environments.
Artun Burak Mecik, M. Yücel, B. Üstündağ· IEEE Access· 0 citations
Multi-step training of sparse, interpretable models of dynamical systems directly from time-series data yields models with accurate short-term dynamics and strong agreement in long-time statistical properties, including mean, variance, and Lyapunov exponents.
The hypothesis is that the sudden transition to an epileptic seizure and the abrupt qualitative change at seizure offset occur through nonlinear processes, while the seizure period is mainly governed by linear autocorrelations, which are responsible for spreading epileptic activity and maintaining ictal dynamics.
David Michel Serrano-Solis, Alberto Isaac Aguilar-Hernández, Wady A. Ríos-Herrera et al.· Brain Topography· 0 citations
Using ordinary least squares regression on high-degree polynomial features with 512-bit arithmetic, a system-agnostic method is introduced that matches the accuracy of standard 64-bit numerical ODE solvers using the systems’ governing equations, suggesting that forecasting low-dimensional chaotic systems from noise-free data is effectively a solved problem.
We here introduce a method to compare time series, leveraging continuous ordinal patterns, i.e., a variant of the classical order pattern analysis, and inspired by a next-generation reservoir computing architecture. The former ones are used as the non-linear transformation units in the latter, in place of traditional neural elements; as a final step, the transformed time series are classified using a standard machine learning model. This allows one to calculate the distance between a time series and a reference, e.g., a randomly shuffled or time-reversed version of the same, and hence to test properties such as stochasticity or time irreversibility. The method is validated through synthetic data generated by classical chaotic systems, demonstrating higher sensitivity when compared to classical tests, and through data representing real-world systems, spanning financial, medical, and technological ones. We finally discuss some additional relevant topics, including the generalizability of the obtained results and the sensitivity of the method to the chosen parameters.
Massimiliano Zanin· Chaos· 0 citations
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