Stability-Aware Parametric Neural ODEs for Bifurcating Dynamical Systems

Parametric dynamical surrogates must recover both the stability of an equilibrium and the nonlinear oscillations that emerge when it loses stability. Near a Hopf bifurcation, weak growth or decay can be difficult to resolve alongside the dominant oscillation, while joint fitting allows nonlinear training to alter the equilibrium spectrum. We introduce a two-stage parametric neural ODE that couples shared operator identification with exact Jacobian preservation. A shared matrix polynomial in the operating parameter is identified jointly across training conditions from normalised early-window trajectory error, with stability penalties, and then frozen. A neural coefficient matrix supplies the nonlinear correction through an amplitude gate. The resulting residual is third order at the equilibrium, so the deployed model’s equilibrium Jacobian is exactly the identified operator at every parameter value, independently of residual training. We evaluate the construction on a thermoacoustic Rijke tube, a cylinder wake and a flexible aircraft wing. The method recovers the correct number of stability crossings on all three systems and accurately locates their boundaries, with errors within the reference resolution on the wake and wing and within a tenth of the grid spacing on the tube. It combines this stability recovery with accurate prediction of nonlinear trajectories and limit-cycle geometry at unseen parameter conditions. Mean per-seed median test trajectory nRMSE is 0.108, 0.169 and 0.147, respectively. Objective comparisons and architectural ablations justify the choices of trajectory fitting, polynomial operator parameterisation, freezing and gating.