Abstract
In this paper, we introduce two approaches to the generalized synchronized synchronization and the inverse generalized synchronization of fractional discrete-time chaotic systems with non-identical dimensions. The convergence of the proposed approaches is established by means of recently developed stability theory. Numerical results are presented based on well-known maps in the literature. Two examples are considered: a 3D generalized synchronization and a 2D inverse generalized synchronization.
Highlights
The theory of fractional calculus is old as its inception can be attributed to two of the most prominent figures of modern calculus, L’Hôpital and Leibniz, as early as 1695
The same cannot be said about discrete fractional calculus, which has not seen the light of day until recently
We aim to propose control laws for two types of synchronization relating to fractional discrete-time systems, namely generalized synchronization (GS) and inverse generalized synchronization (IGS)
Summary
The theory of fractional calculus is old as its inception can be attributed to two of the most prominent figures of modern calculus, L’Hôpital and Leibniz, as early as 1695. The same cannot be said about discrete fractional calculus, which has not seen the light of day until recently. Researchers have found interest in the theory as well as applications of discrete fractional calculus. An exact definition of the fractional discrete operator itself has not yet been agreed upon. The general consensus seems to be restricted to the fact that unlike the integer operator, which is local, the fractional one has infinite memory. This is not at all unlike the Caputo operator in continuous time. It is important to mention that discrete fractional calculus has been shown to provide more accurate models of natural phenomena. The subject has found application in optimal control [3]
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