Fold Bifurcation Identification through Scientific Machine Learning

TL;DR

This work addresses detecting fold bifurcations in dynamical systems from time-series data using a physics-informed CNN. Training on a simple 1-DoF oscillator with nonlinear damping, the network is evaluated on more complex systems (mass-on-moving-belt, van der Pol–Duffing with absorber, and pitch-and-plunge flutter) to test extrapolation capabilities. The key finding is that transforming inputs into polar coordinates and applying a logarithmic amplitude scale, optionally with a moving-mean filter, dramatically improves cross-system performance by suppressing frequency information and highlighting the amplitude decrement toward the fold. The approach shows promise for safety monitoring and real-world applications, though it faces limitations related to noise sensitivity, boundary ambiguity between regions, and the need for broader training data and validation on diverse systems.

Abstract

This study employs scientific machine learning to identify transient time series of dynamical systems near a fold bifurcation of periodic solutions. The unique aspect of this work is that a convolutional neural network (CNN) is trained with a relatively small amount of data and on a single, very simple system, yet it is tested on much more complicated systems. This task requires strong generalization capabilities, which are achieved by incorporating physics-based information. This information is provided through a specific pre-processing of the input data, which includes transformation into polar coordinates, normalization, transformation into the logarithmic scale, and filtering through a moving mean. The results demonstrate that such data pre-processing enables the CNN to grasp the important features related to transient time-series near a fold bifurcation, namely, the trend of the oscillation amplitude, and disregard other characteristics that are not particularly relevant, such as the vibration frequency. The developed CNN was able to correctly classify transient trajectories near a fold for a mass-on-moving-belt system, a van der Pol-Duffing oscillator with an attached tuned mass damper, and a pitch-and-plunge wing profile. The results contribute to the progress towards the development of similar CNNs effective in real-life applications such as safety monitoring of dynamical systems.

Figures (14)

• Figure 1: Illustrative example of a fold bifurcation (red dot) originated by a subcritical Andronov-Hopf bifurcation (blue dot). Green, red and blue colors mark regions far (A), close (B) and after (C) the fold, respectively.
• Figure 2: Bifurcation diagrams of the investigated systems; solid lines: stable solutions, dashed lines: unstable solutions; red dots: fold bifurcations. The lines mark the maximal amplitude of the steady-state solutions for variations of the bifurcation parameter. (a) System in Eq. (\ref{['eq_EOM_NLD']}) for $c_1=0.5$; (b) system in Eq. (\ref{['eq_EoM_MoB']}); (c) system in Eq. (\ref{['eq_EOM_VDP']}) for the parameter values $r=0.05$, $\gamma=0.97$, $\mu_2=0.12$ and $\alpha=0.3$; (d) system in Eq. (\ref{['eq_EOM_PnP']}).
• Figure 3: Mechanical models of the mass-on-moving-belt system (a) and pitch-and-plunge airfoil (b).
• Figure 4: Graphical representation of the utilized CNN architecture.
• Figure 5: Various normalization of the signal. Left column: far from the fold, center column: close to the fold, right column: after the fold. Rows indicate, in order, the original signal, min-max normalization, transformation into polar form and normalization, transformation into polar form and logarithmic scale after normalization. Trajectories were obtained for the system in (\ref{['eq_EOM_NLD']}), with $c_1=0.1$, $c_3=0.0889$, $c_3=0.4222$ and $c_3=0.4889$; $c_{3\text{cr}}=0.444$. The length of the time series in the figure does not strictly reflect the length of the time series used for the training.