Deep learning for centre manifold reduction and stability analysis in nonlinear systems

Bifurcations cause large qualitative and quantitative changes in the dynamics of nonlinear systems with slowly varying parameters. These changes most often are due to modifications that occur in a low-dimensional subspace of the overall system dynamics. The key challenge is to determine what that low-dimensional subspace is, and construct a low-order model that governs the dynamics in that subspace. Centre manifold theory can provide a theoretical means to construct such low-order models for strongly nonlinear systems that undergo bifurcations. Performing a centre manifold analysis, however, is particularly challenging when the system dimensionality is high or impossible when an accurate model of the system is not available. This paper introduces a data-driven approach for identifying a reduced order model of the system based on centre manifold theory. The approach does not require a model of the full order system. Instead, a deep learning approach capable of identifying the centre manifold and the transformation to the centre space is created using measurements of the system dynamics from random perturbations. This approach unravels the characteristics of the system dynamics in the vicinity of bifurcations, providing critical information regarding the behaviour of the system. This article is part of the theme issue 'Data-driven prediction in dynamical systems'.