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Nonlinear systems often require sophisticated approaches for accurate modeling and analysis, with state-space representation being particularly effective. This method is especially useful for systems where variables and parameters vary with time or operating conditions, such as in a simple pendulum or a translational mechanical system with nonlinear springs.
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Deep learning for centre manifold reduction and stability analysis in nonlinear systems.

Amin Ghadami1, Bogdan I Epureanu1

  • 1Department of Mechanical Engineering, University of Michigan, Ann Arbor, MI, USA.

Philosophical Transactions. Series A, Mathematical, Physical, and Engineering Sciences
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PubMed
Summary
This summary is machine-generated.

This study presents a data-driven method to identify low-dimensional models for nonlinear systems near bifurcations. It uses deep learning to find the centre manifold without needing a full system model, improving prediction of system behavior.

Keywords:
bifurcationscentre manifold theorydata-driven methodsnonlinear dynamicsstability

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Area of Science:

  • Dynamical Systems and Nonlinear Dynamics
  • Computational Science and Engineering
  • Data-Driven Modeling

Background:

  • Bifurcations in nonlinear systems cause significant dynamic changes, often driven by low-dimensional subspaces.
  • Centre manifold theory provides a framework for modeling these dynamics but is challenging for high-dimensional or unmodeled systems.

Purpose of the Study:

  • To develop a data-driven approach for identifying reduced-order models based on centre manifold theory.
  • To overcome the limitations of traditional centre manifold analysis for complex and unmodeled systems.

Main Methods:

  • A deep learning approach is employed to identify the centre manifold and its transformation from system dynamics data.
  • The method utilizes measurements from random perturbations to characterize system behavior near bifurcations.

Main Results:

  • Successfully identifies the low-dimensional subspace governing bifurcations without requiring a full system model.
  • Enables the construction of reduced-order models for predicting system dynamics near bifurcations.

Conclusions:

  • The proposed data-driven method effectively unravels system dynamics near bifurcations using deep learning.
  • This approach offers a powerful tool for analyzing and predicting the behavior of complex nonlinear systems.