Year: 2026
Type: article
Abstract: Numerical continuation methods are widely used to follow solution branches of nonlinear systems, but their performance can deteriorate near bifurcation points, where correction iterations may become difficult to achieve convergence and branch-selection issues may arise. In this work, we develop a deep-learning framework for the early detection and classification of approaching bifurcations during Moore–Penrose continuation. A CNN–LSTM model is trained on stochastic time series generated from nonlinear dynamical systems exhibiting saddle-node, Hopf, transcritical, and pitchfork bifurcations. During continuation, stochastic trajectories are generated alongside the computed equilibrium branch using Euler–Maruyama integration, and the resulting time series are supplied to the trained model to obtain bifurcation predictions during the continuation process. Two sampling strategies are investigated to construct these trajectories: continuation-point sampling and adaptive parameter-space sampling. The model achieves an accuracy of 84.4% on the test set and successfully detects bifurcations in several dynamical systems not represented in the training data. The results show that adaptive parameter-space sampling can improve prediction stability when continuation points are irregularly distributed in parameter space. The proposed framework provides an online monitoring mechanism for identifying approaching bifurcations and their predicted class during continuation, thereby providing a basis for future adaptive continuation strategies.
Source: Computers & Structures
Authors: J. Thibodeau, S. Léger
Institution: Université de Moncton
Language: English
Subfield: Statistical and Nonlinear Physics
Field: Physics and Astronomy
Domain: Physical Sciences
Open Access status: hybrid
Estimated APC paid: $4,200
Award: G9001167087
OpenAlex ID: W7218438856