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Clarify the physical process for fractional dynamical systems

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Abstract

Dynamics in fractional order systems has been discussed extensively for presenting a possible guidance in the field of applied mathematics and interdisciplinary science. Within hundreds and thousands of reviews, regular papers and drafts, many fractional differential equations are presented for enjoying mathematical proof without clarifying the scientific background and physical principles. It seems that all nonlinear problems on integer order systems even networks can be confirmed as fractional order systems. This mini-review gives an appropriate clarification on fractional dynamical systems from the physical viewpoint, thereby presenting sufficient evidences for further study on fractional calculus. We argued that non-uniform diffusion, boundary effect and elastic deformation account for the calculation and estimation with fractional order on some physical variables, which can be mapped into dimensionless variables in the dynamical systems. In addition, some similar definitions for energy, wave propagation and diffusion are suggested to find reliable confirmation in the application of fractional calculus.

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Acknowledgements

We would like to give thanks to Dr. Ying Xu and Zhu Zhigang for their editing with the sketches (1, 2), respectively. Dr. Wu Guo-Cheng presented helpful suggestion for preparing a better version.

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  1. School of Science, Chongqing University of Posts and Telecommunications, Chongqing, 430065, China

    Ping Zhou & Jun Ma

  2. Department of Physics, Lanzhou University of Technology, Lanzhou, 730050, China

    Jun Ma

  3. School of Physics, China University of Mining and Technology, Xuzhou, 221116, China

    Jun Tang

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  1. Ping Zhou
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Zhou, P., Ma, J. & Tang, J. Clarify the physical process for fractional dynamical systems. Nonlinear Dyn 100, 2353–2364 (2020). https://doi.org/10.1007/s11071-020-05637-z

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