Emergent Behavior and Computational Capabilities in Nonlinear Systems: Advancing Applications in Time Series Forecasting and Predictive Modeling

Natural dynamical systems can often display various long-term behaviours, ranging from entirely predictable decaying states to unpredictable, chaotic regimes or, more interestingly, highly correlated and intricate states featuring emergent phenomena. That, of course, imposes a level of generality on the models we use to study them. Among those models, coupled oscillators and cellular automata (CA) present a unique opportunity to advance the understanding of complex temporal behaviours because of their conceptual simplicity but very rich dynamics. In this contribution, we review the work completed by our research team over the last few years in the development and application of an alternative information-based characterization scheme to study the emergent behaviour and information handling of nonlinear systems, specifically Adler-type oscillators under different types of coupling: local phase-dependent (LAP) coupling and Kuramoto-like local (LAK) coupling. We thoroughly studied the long-term dynamics of these systems, identifying several distinct dynamic regimes, ranging from periodic to chaotic and complex. The systems were analysed qualitatively and quantitatively, drawing on entropic measures and information theory. Measures such as entropy density (Shannon entropy rate), effective complexity measure, and Lempel–Ziv complexity/information distance were employed. Our analysis revealed similar patterns and behaviours between these systems and CA, which are computationally capable systems, for some specific rules and regimes. These findings further reinforce the argument around computational capabilities in dynamical systems, as understood by information transmission, storage, and generation measures. Furthermore, the edge of chaos hypothesis (EOC) was verified in coupled oscillators systems for specific regions of parameter space, where a sudden increase in effective complexity measure was observed, indicating enhanced information processing capabilities. Given the potential for exploiting this non-anthropocentric computational power, we propose this alternative information-based characterization scheme as a general framework to identify a dynamical system’s proximity to computationally enhanced states. Furthermore, this study advances the understanding of emergent behaviour in nonlinear systems. It explores the potential for leveraging the features of dynamical systems operating at the edge of chaos by coupling them with computationally capable settings within machine learning frameworks, specifically by using them as reservoirs in Echo State Networks (ESNs) for time series forecasting and predictive modeling. This approach aims to enhance the predictive capacity, particularly that of chaotic systems, by utilising EOC systems’ complex, sensitive dynamics as the ESN reservoir.