# Future Challenges in the Fractional-Order Dynamical Systems: From Mathematics to Applications

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Fractional calculus has been known since 1695 when L'Hôpital and Leibniz interchange letters about the noninteger order of the derivative, however, until now, fractional calculus become in a cutting-edge research topic. This renewed interest arises from the fact that fractional calculus preserves the ...

Fractional calculus has been known since 1695 when L'Hôpital and Leibniz interchange letters about the noninteger order of the derivative, however, until now, fractional calculus become in a cutting-edge research topic. This renewed interest arises from the fact that fractional calculus preserves the nonlocality and long memory effects (aka history of all past events), which can be observed in the fields of mathematics, physics, engineering, and so forth. In other words, the fractional-order systems have a memory of the dynamical evolution. As a result, the researchers have proposed novel mathematical models with applications in various fields, such as biology, economics, chaos theory, botany, hidden dynamics, digital circuits, cryptography, control, image processing, wind turbines, viscoelastic studies, ferroelectric materials, and so forth. Those works have demonstrated that fractional derivatives provide an excellent approach for describing elegantly physical real phenomena with improved accuracy.

In this manner, the main purpose of this Research Topic is to cover the remaining research challenges relating not only to the evolution and mathematical foundations of fractional-order calculus, but also to the control, design, and electronic realization of fractional-order systems with an emphasis on engineering applications. Particularly, topics related to general fractional calculus of constant order, general factional calculus of variable order, local fractional calculus, and their extended versions concerning another function are welcome.

The topics to be covered, but not limited, are the following:
• Numerical algorithms with an emphasis on time simulation optimization
• Definitions of variable order derivatives.
• Special functions and applications in fractional calculus.
• New fractional derivative operators.
• Synchronization of fractional-order dynamical systems.
• Fractional-order neural networks
• Fractional-order memristors
• Chaotic dynamics in fractional-order dynamical systems.
• Circuit theory of fractional-order dynamical systems.
• Embedded digital realizations of fractional-order dynamical systems.
• Fractional-order filters and oscillators.
• Control techniques of fractional-order dynamical systems.
• Fractional-order dynamical systems for modeling biological, biochemical, and biomedical phenomena.
• Fractional derivatives in signal processing.
• Stability conditions in fractional-order piecewise-linear dynamical systems
• Semi-analytical solution methods

Keywords: Fractional calculus, Chaotic systems, Discrete systems, Hidden attractors, Memristors, Encryption, Security, Control, Algorithms, FPGAs, electronic circuits.

Important Note: All contributions to this Research Topic must be within the scope of the section and journal to which they are submitted, as defined in their mission statements. Frontiers reserves the right to guide an out-of-scope manuscript to a more suitable section or journal at any stage of peer review.

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