Muhammed Fadera | Searcharxiv

arXiv2026

Global continuation as a complement to traditional continuation and bifurcation analysis

Multistable dynamical systems are ever-prevalent, used to model for example ecosystems, power grids, climate elements, neurons, and more. When perturbed, such systems may ``tip'' from one state of operation to another, often with abrupt, irreversible, and high-impact consequences in each context. Traditionally, these systems are analysed via bifurcation diagrams, the result of a process we refer to as \emph{local continuation}, as it only captures the linear (local) system response to infinitesimal perturbations. Local continuation requires substantial expertise, constant interventions, and may yield inaccurate assessment of the system's response to large perturbations that is crucial for tipping analysis. To address some inherent challenges of local continuation and to provide fundamentally new information during a continuation, this paper introduces \emph{global continuation} as a complement suitable for the study of multistability, critical transitions and real-world-oriented applications. Global continuation finds and continues in parallel (practically) all system attractors and their response to finite perturbations by synthesising information from the whole state space, while placing a focus on the qualities or observables of a dynamical system that the practitioner cares about in context. Global continuation does not require deep expertise and is effortless to use and troubleshoot, making it attractive to applied scientists from different disciplines. We highlight several unique advantages that allow global continuation to complement the status quo and exemplify them through a plethora of representative examples. Global continuation is also implemented as open source software in DynamicalSystems.jl, enhancing its accessibility.

nlin.CD↗

arXiv2025

On the dimension of pullback attractors in recurrent neural networks

Recurrent neural networks trained via the reservoir computing paradigm have demonstrated remarkable success in learning and reconstructing attractors from chaotic systems, often replicating quantities such as Lyapunov exponents and fractal dimensions. It has recently been conjectured that this is because the reservoir computer embeds the dynamics of the chaotic system in its state space before learning. This conjecture has been established for reservoir computers with linear activation functions and remains open for more general reservoir systems. In this work, we employ a non-autonomous dynamical systems approach to establish an upper bound for the box-counting dimension of the pullback attractor, a subset of the reservoir state space that is approximated during training and prediction phases. We prove that the box-counting dimension of the pullback attractor is bounded above by the box-counting dimension of the space of input sequences with respect to the product topology. In particular, for input sequences originating from an Nin-dimensional smooth dynamical system or their generic continuously differentiable observations, the box-counting dimension of the pullback attractor is bounded above by Nin. The results obtained here highlight the fact that, while a reservoir computer may possess a very high-dimensional state space, it exhibits effective low-dimensional dynamics. Our findings also partly explain why reservoir computers are successful in tasks such as attractor reconstruction and the computation of dynamic invariants like Lyapunov exponents and fractal dimensions.

math.DS↗