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Chaotic processes using the two-parameter derivative with non-singular and non-local kernel: Basic theory and
1Department of Mathematical Sciences, University of South Africa, Florida 0003, South Africa.
Chaos (Woodbury, N.Y.)
|September 3, 2016
Summary
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.
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.
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