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A new fractional difference equation two‐dimensional triangle function combination discrete chaotic map (2D‐TFCDM) based on Caputo derivative is proposed. Using the bifurcation diagram, the maximum Lyapunov exponent, and the phase diagram, the numerical solutions of the fractional difference equations are obtained, and the chaotic behavior is observed numerically. After encrypting the key with elliptic curve cryptosystem, the fractional map is developed as an encryption algorithm and applied to color image encryption. Finally, the proposed encryption system is systematically analyzed from five main aspects, and the results show that the proposed encryption system has a good encryption effect. [ABSTRACT FROM AUTHOR]

A kind of top with a fractional operator is discussed in this paper. The top has a periodic nonlinear pulse kick sequence in the magnetic field and constant precessing around the magnetic field. Then, a fractional quantum kicked top map based on the Caputo derivative is proposed. The numerical solutions of the fractional difference equation are obtained, and the chaotic behavior is observed numerically in three aspects. Fractional quantum dynamics behaviors take place in a finite dimensional Hilbert space where the squared angular momentum is free precession. Finally, the dynamic behaviors of the fractional quantum kicked top map are systematically analyzed by using the bifurcation diagram, the phase diagram, and the maximum Lyapunov exponent. [ABSTRACT FROM AUTHOR]

A new fractional two-dimensional quadric polynomial discrete chaotic map (2D-QPDM) with the discrete fractional difference is proposed. Afterwards, the new dynamical behaviors are observed, so that the bifurcation diagrams, the largest Lyapunov exponent plot and the phase portraits of the proposed map are given, respectively. The new discrete fractional map is exploited into color image encryption algorithm and it is illustrated with several examples. The proposed image encryption algorithm is analyzed in six aspects which indicates that the proposed algorithm is superior to other known algorithms as a conclusion. [ABSTRACT FROM AUTHOR]

In this paper, a novel fractional quantum logistic map (FQLM) based on fractional difference is proposed and image encryption algorithm based on it are exploited. First, the FQLM is proposed and its dynamical behaviors are observed with bifurcation diagram, phase portraits and largest Lyapunov exponent plots. After that, the ECC is utilized to cover up the encryption keys and the FQLM is used for the permutation and diffusion process of image encryption. Finally, the cryptosystem is used in image encryption and the algorithm is analyzed in four respects, showing a distinct advantage of the proposed cryptosystem over some of existing ones. [ABSTRACT FROM AUTHOR]

A new fractional two-dimensional triangle function combination discrete chaotic map (2DTFCDM) with the discrete fractional difference is proposed in this paper. The chaos behaviors are observed through the bifurcation diagrams, the largest Lyapunov exponent plot and the phase portraits. The proposed map is applied in color image encryption with the secret keys generated by Menezes–Vanstone Elliptic Curve Cryptosystem. The image encryption system is analyzed using six aspects indicating the superiority of the proposed algorithm compared to the other algorithms. [ABSTRACT FROM AUTHOR]