On Chaos in the Fractional-Order Discrete-Time Unified System and Its Control Synchronization

Abstract

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.

The Co-existence of Different Synchronization Types in Fractional-order Discrete-time Chaotic Systems with Non–identical Dimensions and Orders

“…The authors of [29], again, consider the synchronization of identical fractional Hénon maps. The same can be said regarding [32]. As for [31], the authors propose a simple linear feedback controller suitable for a variety of maps.…”

“…Perhaps the most interesting studies related to the subject are [27][28][29][30][31][32]. In [27], the authors merely consider a pair of identical fractional logistic maps and propose a simple direct synchronization controller.…”

“…With the emergence of fractional chaotic maps such as the fractional Hénon map [25] and the fractional generalized Hénon map [26], the synchronization of such maps became of interest. Very few studies can be found on the subject including [27][28][29][30][31][32].…”

Research Frontier in Chaos Theory and Complex Networks

“…In the paper "On chaos in the fractional-order discrete-time unified system and Its control synchronization", Khennaoui et al [5] introduce a fractional map based on the integer-order unified map and discuss its dynamics and bifurcations. They also propose a one-dimensional adaptive control strategy for forcing the system states towards zero asymptotically, aiming to achieve the complete synchronization of a pair of fractional unified maps with identical or non-identical parameters.…”

The Fractional Form of the Tinkerbell Map Is Chaotic

“…Although fractional maps come with considerable added complexity, they provide better flexibility in the modeling of natural phenomena and lead to richer dynamics with more degrees of freedom. Among the fractional chaotic maps that have been proposed, studied, and applied over the last five years are the fractional logistic map [12], the fractional Hénon map [13], the generalized hyperchaotic Hénon map [14], and the fractional unified map [15]. Perhaps the main concern of the research community has been the possibility of controlling and synchronizing these types of maps [15][16][17][18][19][20].…”