Abstract
In this paper, we propose a fractional map based on the integer-order unified map. The chaotic behavior of the proposed map is analyzed by means of bifurcations plots, and experimental bounds are placed on the parameters and fractional order. Different control laws are proposed to force the states to zero asymptotically and to achieve the complete synchronization of a pair of fractional unified maps with identical or nonidentical parameters. Numerical results are used throughout the paper to illustrate the findings.
Highlights
Chaotic dynamical systems have attracted a considerable level of attention over the last three decades due to the wide range of applications
Since the subject of fractional discrete calculus is still relatively new and the notation has not yet been settled, we start with a general description of the notation and stability results that will aid the reader in understanding the analysis to come
We note that similar to continuous-time fractional calculus, where numerous definitions exist for the fractional derivative of a function, the Caputo one is the most used today
Summary
Chaotic dynamical systems have attracted a considerable level of attention over the last three decades due to the wide range of applications. Since the Henon and Lozi maps are the earliest discrete-time chaotic systems, they have been studied extensively by a vast number of researchers. One of the interesting studies is that of Zeraoulia and Sprott [8], where the authors proposed a new chaotic map as a combination of the the Henon and Lozi maps named the unified map. In [19], the authors point out that the chaotic patterns exhibited by the fractional generalized Henon map depend on the fractional order This means that the fractional map is more suitable for secure communications and encryption, as it includes a new degree of freedom. To the best of our knowledge, very few studies have been dedicated to the control and synchronization of fractional-order chaotic maps, including [33,34,35]
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