Bifurcation curve detection with deflation for multiparametric PDEs

TL;DR

The paper tackles the challenge of identifying bifurcation structures in nonlinear multiparametric PDEs where multiple solution branches coexist. It introduces a deflated arclength continuation framework that extends arclength methods to multi-parameter spaces and couples them with a zigzag path-following strategy to robustly locate bifurcation curves and surfaces. The key contributions are the deflated arclength algorithm, the multiparametric extension with explicit tangent and path formulations, and the zigzag detector for curve/ surface localization, all validated on Bratu and Allen-Cahn benchmarks across 1D/2D domains and for parameter counts up to $p=3$. The methods demonstrate path-independence, accuracy in reconstructing full diagrams, and effective detection of bifurcation regions without prior knowledge of their type, with potential for integration into reduced-order models for large-scale multiparametric PDE analyses.

Abstract

This work presents a comprehensive framework for capturing bifurcating phenomena and detecting bifurcation curves in nonlinear multiparametric partial differential equations, where the system exhibits multiple coexisting solutions for given values of the parameters. Traditional continuation methods for one-dimensional parameterizations employ the previously computed solution as the initial guess for the next parameter value. These are usually very inefficient, since small step sizes increase computational cost, while larger steps could jeopardize the method convergence jumping to a different solution branch or missing the bifurcation point. To address these challenges, we propose a novel framework that combines: (i) arclength continuation, adaptively selecting new parameter values in higher dimension, and (ii) the deflation technique, discovering multiple branches to construct complete bifurcation diagrams. In particular, the arclength continuation method is designed to handle multiparametric scenarios, where the parameter vector $λ\in \mathbb{R}^p$ traces a curve $g(λ)$ within a $p$-dimensional parameter space. In addition, we introduce a zigzag path-following strategy to robustly track the bifurcation curves and surfaces, respectively, for two- and three-dimensional parametric spaces. Finally, we demonstrate its performance on two benchmark problems: the Bratu equation and the Allen-Cahn equation.

Figures (17)

• Figure 1: Visualization of the arclength continuation method.
• Figure 2: Sketch of pitchfork bifurcation discovery.
• Figure 3: Sketch of saddle-node bifurcation discovery.
• Figure 4: Paths for the multiparametric arclength continuation with $a=1$, $b=10$, $c=1$, and $d=5$.
• Figure 5: Bifurcation curve and the approximating zigzag tracking path (red) on the $(\lambda_1,\lambda_2)$-domain with $p=2$ with the zoom-in showing the local bifurcation diagram.

Theorems & Definitions (2)

• Remark
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