Adjoint-based Hopf-bifurcation Instability Suppression via First Lyapunov Coefficient

TL;DR

This work addresses the challenge of integrating Hopf-bifurcation stability into gradient-based design optimization by leveraging the first Lyapunov coefficient as a local stability metric. It develops an efficient adjoint-based (RAD) formulation to compute derivatives of the first Lyapunov coefficient, deriving a two-branch adjoint system and a RAD expression that is independent of the number of design variables. The authors demonstrate the approach on three problems—a manufactured algebraic Hopf model, a nonlinear aeroelastic typical-section, and a complex Ginzburg–Landau PDE—where subcritical bifurcations are suppressed and stable, supercritical LCOs are obtained. The method promises scalable bifurcation-constrained optimization for aeroelastic/aerodynamic design and other PDE-governed systems, with potential extensions to multi-mode Hopf phenomena and larger nonlinear setups.

Abstract

Many physical systems exhibit limit cycle oscillations induced by Hopf bifurcations. In aerospace engineering, limit cycle oscillations arise from undesirable Hopf bifurcation phenomena such as aeroelastic flutter and transonic buffet. In some cases, the resulting limit cycle oscillations can themselves be unstable, leading to amplitude divergence or hysteretic transitions that threaten structural integrity and performance. Avoiding such phenomena when performing gradient based design optimization requires a constraint that quantifies the stability of the bifurcations and the derivative of that constraint with respect to the design variables. To capture the local stability of bifurcations, we leverage the first Lyapunov coefficient, which predicts whether the resulting limit cycle oscillation is stable or unstable. We develop an accurate and efficient method for computing derivatives of the first Lyapunov coefficient. We leverage the adjoint method and reverse algorithmic differentiation to efficiently compute the derivative of the first Lyapunov coefficient. We demonstrate the efficacy of the proposed adjoint method in three design optimization problems that suppress unstable bifurcation: an algebraic Hopf bifurcation model, an aeroelastic model of a typical section, and a nonlinear problem based on the complex Ginzburg-Landau partial differential equation. While the current formulation addresses only a single bifurcation mode, the proposed adjoint shows great potential for efficiently handling Hopf bifurcation constraints in large scale nonlinear problems governed by partial differential equations. Its accuracy, versatility and scalability make it a promising tool for aeroelastic and aerodynamic design optimization as well as other engineering problems involving Hopf bifurcation instabilities.

Figures (13)

• Figure 1: Stable, unstable, and semi-stable Hopf bifurcations. For each case, the motion magnitude, $h$, approaches the limit $h \rightarrow 0$ to measure the bifurcation stability at the equilibrium point. The direction of motion for solutions in the vicinity of the limit cycle oscillation (LCO) is indicated by the arrows.
• Figure 2: Three levels of stability analysis related to Hopf bifurcation: linear, Hopf bifurcation, and LCO stability. The proposed method builds on the first two columns.
• Figure 3: XDSM for the bifurcation stability analysis and optimization.
• Figure 4: Contours of the stability metric, $f_{\text{Lyp}}$ (left), and the bifurcation parameter, $\mu$, (right). The brown arrows indicate the path of the optimization.
• Figure 5: Optimization history by major iteration showing response going from subcritical to supercritical. The color indicates the iteration index.

Theorems & Definitions (2)

• Theorem IV.1
• proof