Discrete tempered fractional calculus for new chaotic systems with short memory and image encryption

Fractional derivatives with memory effects have been widely used in image processing. This study investigates a discrete analogy of tempered fractional calculus on an isolated time scale and provides a new kind of discrete fractional calculus. Some useful properties and discrete Mittag–Leffler functions are derived. Numerical formulae are obtained accordingly. Finally, a new fractional logistic map with two parameters is proposed and chaotic behavior is illustrated. Image encryption is given as the application to show high security.