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The study introduces a new two-parameter fractional derivative, revealing the second parameter β of the Mittag-Leffler function influences chaotic solutions by twisting and rotating them, not interrupting the chaos itself.

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Area of Science:

  • Fractional Calculus
  • Mathematical Modeling
  • Non-linear Dynamics

Background:

  • Recent advancements addressed singularity and locality in mathematical modeling.
  • The influence of the second parameter (β) of the two-parameter Mittag-Leffler function (Eα,β(z)) remained an open question.

Purpose of the Study:

  • To develop a two-parameter fractional derivative with a non-singular and non-local kernel.
  • To investigate the impact of the parameter β on natural phenomena modeling.
  • To establish the mathematical framework for analyzing two-parameter fractional differential equations.

Main Methods:

  • Generalization of a one-parameter fractional derivative to a two-parameter version based on Eα,β(z).
  • Utilization of Agarwal/Erdelyi higher transcendental functions and their Laplace transforms.
  • Application to the Lorenz system for atmospheric convection modeling, including existence and numerical schemes.

Main Results:

  • Explicit expressions for the Laplace transforms of the two-parameter derivatives and the associated fractional integral were derived.
  • Chaotic behaviors were observed in solutions for α < 0.55.
  • The parameter β was found to indirectly influence solutions by squeezing and rotating them, creating a twisting effect without interrupting the chaos.

Conclusions:

  • The second parameter β of the two-parameter Mittag-Leffler function has a substantial impact on modeling natural phenomena.
  • The newly developed two-parameter fractional derivative offers new possibilities for advanced mathematical modeling.
  • The observed twisting and rotation effects of parameter β open new avenues for understanding complex system dynamics.