A new Rabotnov fractional‐exponential function‐based fractional derivative for diffusion equation under external force

Variable-order fractional discrete-time recurrent neural networks

A new fractional-order hyperchaotic memristor oscillator: Dynamic analysis, robust adaptive synchronization, and its application to voice encryption

Discrete fractional calculus ([Formula: see text]) is significant for neural networks, complex dynamic systems and frequency response analysis approaches. In contrast with the continuous-time frameworks, fewer outcomes are accessible for discrete fractional operators. This study investigates some major consequences of two sorts of inequalities by considering discrete Atangana–Baleanu [Formula: see text]-fractional operator having [Formula: see text]-discrete generalized Mittag-Leffler kernels in the sense of Riemann type ([Formula: see text]). Certain novel versions of reverse Minkowski and related Hölder-type inequalities via discrete [Formula: see text]-fractional operators having [Formula: see text]-discrete generalized Mittag-Leffler kernels are given. Moreover, several other generalizations can be generated for nabla [Formula: see text]-fractional sums. The proposing discretization is a novel form of the existing operators that can be provoked by some intriguing features of chaotic systems to design efficient dynamics description in short time domains. Furthermore, by combining two mechanisms, numerous new special cases are introduced. 

SOME RECENT DEVELOPMENTS ON DYNAMICAL ℏ-DISCRETE FRACTIONAL TYPE INEQUALITIES IN THE FRAME OF NONSINGULAR AND NONLOCAL KERNELS

Laser chaotic systems have been extensively studied and applied in the past 20 years. However, the fractional form of the chaotic system of erbium-doped fiber laser has not been studied yet. Therefore, this paper proposes a fractional-order double-ring erbium-doped fiber laser chaotic system. The dynamic characteristics of the system are studied by phase diagram, bifurcation diagram, Lyapunov exponent spectrum, Spectral Entropy (SE) complexity, and coexisting attractors. At the same time, the digital circuit of the new system is implemented on the digital signal processing (DSP) platform. Compared with integer-order double-ring erbium-doped fiber laser chaotic systems, this fractional-order chaotic system has more complex dynamics and higher sensitivity and randomness. Finally, the fractional-order double-ring erbium-doped fiber laser chaotic system is applied to the image encryption algorithm with the improved gravitation model. In this encryption algorithm, the diffusion part uses an improved universal gravitation model, and the Zigzag algorithm scrambles the pixels. The security of the algorithm is analyzed. The analyses of the experimental results show that the image encryption scheme proposed in this paper has a good encryption effect and good security performance. The research in this paper provides theoretical guidance and experimental basis for the research of fractional-order laser chaotic secure communication.

Fractional-order double-ring erbium-doped fiber laser chaotic system and its application on image encryption

New variable-order fractional chaotic systems are proposed in this paper. A concept of short memory is introduced where the initial point in the Caputo derivative is varied. The fractional order is defined by the use of a piecewise constant function which leads to rich chaotic dynamics. The predictor-corrector method is adopted, and numerical solutions of fractional delay equations are obtained. Then, this concept is extended to fractional difference equations, and generalized chaotic behaviors are discussed numerically. Finally, the new fractional chaotic models are applied to block image encryption and each block has a different fractional order. The new chaotic system improves security of the image encryption and saves the encryption time greatly.

De-Qiang Zeng

Papers

New variable-order fractional chaotic systems for fast image encryption.