Parametric Spectral Submanifolds across Hopf Bifurcations with Applications to Fluid Dynamics

TL;DR

This paper studies how spectral submanifolds behave near Hopf bifurcations in high-dimensional systems, providing theoretical insights and practical models for fluid dynamics transitions.

Contribution

It offers a new analysis of SSM persistence and smoothness near bifurcations, including resonance effects, with applications to fluid flow models.

Findings

Low-order SSM coefficients persist through bifurcation

Resonances limit SSM smoothness but can be managed

Parametric SSM models accurately predict fluid flow transitions

Abstract

We investigate the persistence and regularity of spectral submanifolds (SSMs) in high-dimensional parametric dynamical systems undergoing a Hopf bifurcation. By analyzing how resonances in the linearized spectrum near bifurcation points limit the existence and smoothness of SSMs, a phenomenon that has been mostly overlooked, we show that low-order Taylor coefficients of the SSM expansion and the associated reduced dynamics persist smoothly through the bifurcation. This analysis generalizes to any local bifurcation and provides a clear estimate of the parameter ranges over which a parametric SSM model can be justified, thus illustrating how globally the model can be extended despite the presence of resonances near criticality. We demonstrate these findings on multiple examples, including a data-driven SSM approach to the lid-driven cavity flow. For that problem, we construct a parametric SSM-reduced model that accurately captures the full transition to periodic dynamics and the critical Reynolds number. These results provide a mathematical foundation for robust data- and equation-driven model reduction of fluid flows across bifurcations, enabling an accurate prediction of nonlinear dynamics across critical parameter regimes.