Bifurcation curve detection with deflation for multiparametric PDEs

TL;DR

This paper introduces a novel framework combining arclength continuation and deflation techniques to efficiently detect bifurcation curves in nonlinear multiparametric PDEs, improving the robustness and completeness of bifurcation analysis.

Contribution

It presents a new approach that integrates arclength continuation with deflation for multiparametric PDEs, enabling comprehensive bifurcation curve detection in higher-dimensional parameter spaces.

Findings

Successfully applied to Bratu and Allen-Cahn equations

Robustly tracks bifurcation surfaces in 2D and 3D parameter spaces

Demonstrates improved efficiency over traditional methods

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 $\lambda \in \mathbb{R}^p$ traces a curve $g(\lambda)$ 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.