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Adaptive stochastic continuation with a modified lifting procedure applied to complex systems.

Clemens Willers1,2, Uwe Thiele1,2,3, Andrew J Archer4,5

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Summary
This summary is machine-generated.

This study enhances stochastic continuation methods for analyzing complex systems by adaptively choosing algorithm parameters and introducing lifting techniques. This improves the accuracy of bifurcation diagrams, especially near critical points.

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Area of Science:

  • Complex Systems Analysis
  • Computational Physics
  • Statistical Mechanics

Background:

  • Complex systems in natural and social sciences are often modeled using microscopic approaches like lattice- or agent-based models.
  • Analyzing macroscopic observables and bifurcation structures of these systems requires equation-free methods, such as stochastic continuation.
  • Existing stochastic continuation methods face challenges in accurately capturing system dynamics and bifurcations.

Purpose of the Study:

  • To improve the accuracy and reliability of stochastic continuation techniques for analyzing complex systems.
  • To develop adaptive parameter selection and lifting techniques for enhanced bifurcation analysis.
  • To provide a robust method for calculating macroscopic observables and their statistical errors.

Main Methods:

  • Adaptive parameter selection for stochastic continuation algorithms.
  • Introduction of lifting techniques to generate microscopic states with natural structure.
  • Calculation of fixed points for fluctuating functions using linear fits to estimate statistical error.
  • Application to diverse models: 2D Ising model, active Ising model, and stochastic Swift-Hohenberg model.

Main Results:

  • Significantly improved accuracy in obtaining bifurcation diagrams, particularly near bifurcation points.
  • Demonstrated reliability of lifting techniques for evaluating macroscopic quantities.
  • Established a simple yet effective method for measuring statistical error in fluctuating functions.
  • Successful application and validation across multiple complex system models.

Conclusions:

  • The enhanced stochastic continuation approach offers a powerful tool for analyzing complex systems.
  • Adaptive parameter choice and lifting techniques are crucial for accurate bifurcation analysis.
  • The method provides a reliable way to assess statistical errors, enhancing model interpretability.
  • Further research is needed to address remaining challenges and expand the technique's applicability.