Critical transitions, abrupt and significant shifts in system dynamics, are observed across diverse fields, including physiology (heart rhythm changes), economics (market crashes), and ecology (ecosystem collapses). Bifurcation theory, focusing on sudden qualitative changes in dynamical systems as parameters change, provides a framework for understanding these transitions. Many bifurcations exhibit critical slowing down—reduced system stability—manifested in time series properties like variance and autocorrelation. These properties serve as early warning signals (EWS) but have limitations in predicting bifurcation type and failing in systems with nonsmooth potentials or noise-induced transitions. Deep learning offers a promising alternative, leveraging the universal properties of bifurcations to learn generic features from simulated data, thereby providing EWS without requiring extensive data from the specific system under study. Previous work successfully applied deep learning to continuous-time bifurcations; however, the unique dynamics of discrete-time bifurcations remain largely unexplored in this context. This study addresses this gap by developing and evaluating a deep learning classifier to provide specific EWS for discrete-time bifurcations, aiming for improved accuracy and the identification of bifurcation type.

Existing early warning signals (EWS) for critical transitions often rely on analyzing changes in time series properties like variance and autocorrelation. While these indicators can detect critical slowing down, a hallmark of bifurcations, they often struggle to accurately predict the *type* of bifurcation that's approaching. Analytical approximations exist for these properties near different bifurcations, but their effectiveness is limited by factors like noise intensity and the rate of approach to the bifurcation. Recent advancements in deep learning have offered a potential solution by training neural networks to classify time series based on the approaching bifurcation type. This approach leverages the universal properties of bifurcations, allowing for accurate prediction even with limited data from the target system. However, most previous deep learning approaches focused on *continuous-time* bifurcations, neglecting the distinct characteristics of discrete-time systems.

This study trains a deep learning classifier to predict five local codimension-one discrete-time bifurcations: period-doubling, Neimark–Sacker, fold, transcritical, and pitchfork. The training data consists of 50,000 simulated trajectories generated from normal form equations (with higher-order terms and noise) representing each bifurcation type. The classifier uses a CNN-LSTM architecture. Two classifiers are trained independently: Classifier 1 uses data from the middle of the time series, providing an early warning; Classifier 2 uses data closer to the bifurcation, improving specificity. The ensemble prediction of both classifiers is utilized for final predictions. The classifier's performance is evaluated on withheld test data and compared against the performance of variance and lag-1 autocorrelation as EWS. The evaluation includes simulations from five discrete-time models (Fox, Westerhoff, Ricker, Lotka-Volterra, and Lorenz models) across varied noise intensities and rates of forcing. Furthermore, experimental data from spontaneously beating chick-heart aggregates undergoing period-doubling bifurcations are used for real-world validation. The performance of the EWS is assessed using the F1 score (combining sensitivity and specificity) and the area under the ROC curve (AUC).

The deep learning classifier demonstrates superior performance compared to traditional EWS (variance and lag-1 autocorrelation) across various models and noise levels. On withheld test data, Classifier 1 achieved an F1 score of 0.66, while Classifier 2 achieved 0.85, showcasing the improved specificity of using data closer to the bifurcation. For binary classification (bifurcation or not), F1 scores reached 0.79 and 0.97 respectively. The classifier accurately identified period-doubling, Neimark–Sacker, and fold bifurcations with high sensitivity and specificity. The transcritical and pitchfork bifurcations were sometimes confused, likely due to their similar normal forms. Across the five theoretical models, the classifier consistently outperformed variance and lag-1 autocorrelation in terms of AUC score, particularly in scenarios with lower rates of forcing. The classifier's ability to predict the correct bifurcation type was especially high for period-doubling and Neimark–Sacker bifurcations. In the chick-heart aggregate experiments, the classifier correctly predicted period-doubling in 16 out of 23 cases, exhibiting higher AUC than variance while also providing bifurcation type information. The performance was robust to different smoothing methods and parameters.

The study demonstrates the efficacy of deep learning in predicting discrete-time bifurcations, offering a more reliable and informative EWS than traditional methods. The classifier's superior performance across diverse models and noise levels highlights its potential for real-world applications. Its ability to predict the type of bifurcation is crucial, as different bifurcations lead to qualitatively different post-transition dynamics. Accurate prediction of bifurcation type could be invaluable in preventing harmful transitions (e.g., dangerous heart rhythms) or promoting beneficial ones (e.g., ecosystem recovery). The results support the integration of dynamical systems and deep learning methodologies for early warning systems.

This research establishes the effectiveness of a deep learning classifier for predicting discrete-time bifurcations, surpassing traditional EWS in accuracy and providing crucial information on bifurcation type. Future research could explore alternative neural network architectures, hyperparameter optimization, and hierarchical classification approaches to further improve performance. Expanding the training data to include more diverse null trajectories and exploring a unified classifier for both continuous and discrete-time bifurcations would also be valuable.

The classifier's current design is limited to predicting a specific subset of discrete-time bifurcations (local, codimension-one). The accuracy in distinguishing between transcritical and pitchfork bifurcations could be improved. While the study tested robustness to noise and forcing rates, real-world systems may present additional complexities that warrant further investigation. The definition of 'null' trajectories could be refined for enhanced training data.