Modern Dynamical Systems and Applications

Dear Colleagues,

Dynamical systems theory provides one of the most versatile languages available to applied mathematics, describing how physical, biological, economic and social systems evolve in time. In recent years, the field has been reshaped by the availability of large volumes of observational data and by machine learning methods able to exploit them—from reservoir computing and next-generation reservoir computing (NG-RC) to physics-informed neural networks (PINNs), neural operators, transformer architectures, and emerging foundation models capable of zero-shot forecasting. This has opened a genuinely two-way exchange: data-driven techniques now support model discovery, parameter estimation, forecasting, and the detection of bifurcations, synchronisation and chaos, while concepts from nonlinear dynamics—stability, attractors, invariant manifolds, Lyapunov exponents, Koopman operator theory, and delay embedding theorems—are increasingly used to understand the behaviour of learning algorithms themselves. Methodological tools such as sparse identification of nonlinear dynamics (SINDy), recurrence quantification analysis, information-theoretic measures (transfer entropy, mutual information), and fractal dimension estimation further bridge the gap between first-principles modelling and data-driven inference.

This Special Issue seeks original research articles and high-quality reviews at this interface. Contributions are welcome on the analytical and numerical study of continuous and discrete dynamical systems, chaotic and hyper-chaotic behaviour, complex networks and emergent collective phenomena, as well as on machine learning approaches to the modelling, control and prediction of nonlinear systems. Topics of particular interest include, but are not limited to: reservoir computing and its next-generation variants for chaotic time series; physics-informed and physics-guided learning strategies; neural operators and transformer-based architectures for dynamics; foundation models for universal forecasting; data-driven bifurcation detection and critical transition prediction; information-theoretic approaches to nonlinear causality; and hybrid methods combining dynamical invariants with deep learning. Rigorous applications in physics, engineering, mathematical biology, energy, economics and the social sciences are equally encouraged. Both theoretical advances and well-founded methodological contributions will be considered.

We look forward to receiving your contributions.

Dr. David Becerra-Alonso
Guest Editor

Manuscript Submission Information

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Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2600 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

• nonlinear dynamical systems and chaos
• bifurcation, stability and synchronisation
• reservoir computing and next-generation reservoir computing
• physics-informed neural networks and neural operators
• machine learning and foundation models for dynamics
• nonlinear time series forecasting and data-driven model discovery
• Koopman operator theory and delay embedding
• complex networks and emergent collective behaviour
• information-theoretic methods in nonlinear dynamics
• sparse identification and hybrid physics-ML approaches