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Parametric Spectral Submanifolds across Hopf Bifurcations with Applications to Fluid Dynamics

James King, Bálint Kaszás, Gergely Buza, William Jussiau, George Haller

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.

Parametric Spectral Submanifolds across Hopf Bifurcations with Applications to Fluid Dynamics

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.
Paper Structure (27 sections, 5 theorems, 82 equations, 4 figures, 2 tables)

This paper contains 27 sections, 5 theorems, 82 equations, 4 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Prior Work
  3. Spectral Submanifolds (SSMs)
  4. Continuing SSMs through a Hopf bifurcation
  5. Invariant manifolds near a Hopf Bifurcation
  6. Accumulation of resonances near a Hopf bifurcation
  7. Existence of invariant manifolds at resonance
  8. Summary of invariant manifolds near a Hopf Bifurcation
  9. Persistence of low-order SSM coefficients through Hopf bifurcation
  10. Results
  11. Parametric SSMs for the Hopf Normal Form
  12. Data-Driven Parametric SSMs for the Lid-Driven Cavity Flow
  13. Setup
  14. Parametric SSM construction
  15. Data generation and Training
  16. ...and 12 more sections

Key Result

Lemma 3.1

Under the assumptions assump:bifurcating_eigenvalue_pair-assump:simple_disjoint_eigvals, the nonresonance conditions eq:nonresonance_conditions_3D cannot hold uniformly on any open neighborhood of $\mu_0$.

Figures (4)

  • Figure 1: Analytic and fractional SSMs of system \ref{['eq:toy_model_hopf']} with trajectories overlayed in (a)-(c), and as a cross-section in (d).
  • Figure 2: Distribution of training and testing parameters.
  • Figure 3: Comparison of the SSM-reduced dynamics and SSMs at different previously unseen Reynolds numbers. The shift mode was introduced in noackHierarchyLowdimensionalModels2003noackHierarchyLowdimensionalModels2003
  • Figure 4: Parametric SSM predictions for the lid-driven cavity flow at previously unseen Reynolds number.

Theorems & Definitions (12)

  • Lemma 3.1: Failure of uniform nonresonance near Hopf bifurcation
  • proof
  • Lemma 3.2: Asymptotic localization of resonances near a Hopf bifurcation
  • proof
  • Remark 3.3
  • Lemma 3.4: Persistence of low-order SSM coefficients
  • proof
  • Lemma 3.5: Persistence of low-order reduced dynamics coefficients
  • proof
  • Theorem 3.6: Smooth computation of low-order SSM coefficients through a Hopf bifurcation
  • ...and 2 more