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An improved cell mapping method based on dimension-extension for fractional systems.

Minjuan Yuan1, Liang Wang1, Yiyu Jiao1

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This study introduces an improved cell mapping method for analyzing fractional systems, overcoming memory limitations. The new approach accurately captures complex dynamics without losing information, enabling better understanding of fractional oscillators.

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Area of Science:

  • Dynamical Systems
  • Nonlinear Dynamics
  • Fractional Calculus

Background:

  • Analyzing global dynamics of fractional systems is difficult due to their inherent memory property.
  • The standard cell mapping method is not directly applicable to fractional systems without the Markov assumption.

Purpose of the Study:

  • To develop an improved cell mapping method for investigating the global dynamics of fractional systems.
  • To overcome the limitations of existing methods in handling the memory effects of fractional systems.

Main Methods:

  • An improved cell mapping method based on dimension-extension is proposed.
  • Additional auxiliary variables are introduced to localize the nonlocal problem in a higher dimension space.
  • The evolution process is modeled using Markov chains in the extended space.

Main Results:

  • The proposed method successfully describes the one-step mappings of fractional systems as Markov chains.
  • Global dynamics of fractional systems are obtained accurately and efficiently without memory loss.
  • Abundant global dynamic behaviors were observed in fractional smooth and discontinuous oscillators.

Conclusions:

  • The dimension-extension based cell mapping method is effective for analyzing the global dynamics of fractional systems.
  • This approach provides a robust framework for studying complex behaviors in fractional oscillators.
  • The method demonstrates high accuracy and efficiency in simulations.