A fractal-fractional gingerbread-man map generalized by ρ-fractal-fractional difference operator

Abstract

By using the generalization of the gamma function (ρ-gamma function: Гρ(.)), we introduce a generalization of the fractal-fractional calculus which is called ρ-fractal-fractional calculus. Examples are illustrated including the basic power functions. As applications, we formulate the ρ-fractal-fractional difference operators. A class of maps, called gingerbread-man maps, is investigated. We present a new idea of a stability for continuous system, based on three parameters. Sufficient conditions are illustrated to obtain the stability of the system.