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Caputo-Hadamard fractional differential equations on time scales: Numerical scheme, asymptotic stability, and chaos.

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Data-driven discrete fractional chaotic systems, new numerical schemes and deep learning.

Guo-Cheng Wu1, Zhi-Qiang Wu1, Wei Zhu1

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This study introduces novel numerical schemes for fractional chaotic systems, enabling accurate parameter estimation using deep learning. The findings highlight an efficient method for discrete-time fractional systems.

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Area of Science:

  • Applied Mathematics
  • Computational Science
  • Chaos Theory

Background:

  • Parameter estimation is crucial for data-driven fractional chaotic systems.
  • Discretization of fractional calculus operators presents significant challenges, limiting research.
  • Existing methods for parameter estimation in these systems are underdeveloped.

Purpose of the Study:

  • To develop new numerical schemes for discretizing delay fractional difference equations.
  • To estimate unknown parameters in discrete fractional chaotic systems using neural networks.
  • To provide a robust and accurate deep learning-based method for parameter estimation.

Main Methods:

  • Derivation of novel numerical schemes for Caputo and Riemann-Liouville fractional difference equations.
  • Construction of appropriate loss functions for parameter estimation.
  • Application of a neural network approach for estimating system parameters.
  • Robustness analysis under varying noise levels.

Main Results:

  • Newly derived numerical schemes enable accurate discretization of fractional operators.
  • The neural network method achieved high accuracy in parameter estimation compared to true values.
  • The proposed method demonstrated robustness under different noise conditions.
  • Validation of the efficiency of the deep learning approach for fractional discrete-time systems.

Conclusions:

  • The paper presents an efficient deep learning method for parameter estimation in discrete-time fractional chaotic systems.
  • The developed numerical schemes overcome challenges in discretizing fractional calculus.
  • The findings offer a valuable contribution to the field of data-driven fractional dynamics.