P-Bifurcations in Stochastic Flutter Model Under Common Gust Perturbations

Abstract

Aeroelastic flutter represents a critical nonlinear instability in flight dynamics, where the coupling between structural elasticity and unsteady aerodynamics leads to self-excited oscillations. In deterministic settings, the onset of flutter is typically characterized by bifurcations of invariant sets such as equilibria or limit cycles. However, real flight conditions are inherently stochastic due to atmospheric turbulence, rendering trajectory-based attractors insufficient for describing long-time behavior and motivating a probabilistic viewpoint. The stochastic nature of turbulence modifies these transitions, often generating high-dimensional stationary distributions which are difficult to visualize. In this work, we use a topological framework to detect and characterize such stochastic bifurcations in a two-degree-of-freedom aerofoil model with nonlinear stiffness. Reconstructing the full phase-space kernel density estimate (KDE) and constructing homological bifurcation plots reveal high-dimensional toroidal structures in the stationary probability density that are otherwise difficult to detect from two-dimensional projections. Further, we perform a comparative analysis of flutter under the influence of three classes of gust models: sinusoidal white Gaussian noise, the Dryden turbulence model, and the Von Karman turbulence model. Our analysis bypasses the predominantly used visual inspection in stochastic bifurcation studies, enabling systematic and automated exploration of stochastic flutter across large parameter ranges.

Figures (7)

• Figure 1: 2-DOF aerofoil model with pitch and plunge motion.
• Figure 2: Stochastic gust excitation models considered in this study: sinusoidal, Dryden, and Von Karman. The top row shows the probability distributions of the stochastic velocity fluctuations $u(t)$ generated by each excitation model, while the bottom row shows the corresponding distributions of the resulting flow speed $U(t)$ used in the aeroelastic simulations.
• Figure 3: Schematic overview of the homological bifurcation analysis framework. Monte Carlo simulations generate ensembles of trajectories which are used to estimate the distribution. Superlevel filtration of cubical complexes is computed for each KDE, and the resulting Betti vectors yield the homological bifurcation plot,
• Figure 4: Representative time-series of pitch $\alpha(t)$ and plunge $\epsilon(t)$ with their velocities, at subcritical, near-critical, and supercritical flow speeds for the sinusoidal, Dryden, and Von Karman excitation models.
• Figure 5: Representative phase-space projections $(\alpha,\varepsilon)$, $(\varepsilon,\dot{\varepsilon})$ and $(\alpha,\dot{\alpha})$ showing limit cycles for the three noise models at a speed of 10 m/s. The limit cycles are very narrow in $(\alpha,\varepsilon)$ space, but wider in the other two.