Manuscript Topics

Recent advances in nonlinear dynamics have emerged from an increasingly interdisciplinary landscape that combines sophisticated analytical frameworks with state-of-the-art computational methodologies. From an analytical standpoint, developments such as extended continuation techniques, multiscale decomposition methods, and geometric normal-form theory have enhanced the capacity to investigate strongly nonlinear and high-dimensional systems. These approaches have been further strengthened by substantial progress in computational ergodic theory, which provides rigorous tools for the statistical characterisation of strange attractors and other complex invariant structures.

On the computational front, the integration of adaptive numerical schemes with machine-learning mechanisms — notably Neural Ordinary Differential Equations (Neural ODEs) and Physics-Informed Neural Networks (PINNs) — has driven a paradigm shift in the reconstruction of dynamical fields, the forecasting of chaotic phenomena, and parameter inference under conditions of partial observability. Enhanced computational capabilities, enabled by parallel algorithms optimised for GPUs and distributed architectures, now support simulations at unprecedented scales. Additionally, the adoption of topological methods, particularly Topological Data Analysis (TDA), has introduced new avenues for identifying structural invariants in nonlinear time series, offering insights beyond those accessible through classical techniques.

Collectively, these coordinated analytical and computational developments substantially expand the capacity to model, predict, and control complex nonlinear systems, with growing relevance across diverse domains including climate science, engineering, biology, neuroscience, and robotics.

Keywords: nonlinear systems; bifurcation theory; computational ergodic theory; complex networks; neural ordinary differential equations; physics-informed neural networks; biosystem and ecosystem dynamics; social dynamics

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