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

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.