Abstract
In this paper, we propose three fractional chaotic maps based on the well known 3D Stefanski, Rössler, and Wang maps. The dynamics of the proposed fractional maps are investigated experimentally by means of phase portraits, bifurcation diagrams, and Lyapunov exponents. In addition, three control laws are introduced for these fractional maps and the convergence of the controlled states towards zero is guaranteed by means of the stability theory of linear fractional discrete systems. Furthermore, a combined synchronization scheme is introduced whereby the fractional Rössler map is considered as a drive system with the response system being a combination of the remaining two maps. Numerical results are presented throughout the paper to illustrate the findings.
Highlights
Chaotic discrete-time systems have received considerable attention over the last two decades due to their many applications in secure communications [1,2,3,4] and control [5]
Fractional maps usually exhibit a chaotic attractor over a range of fractional orders, which increases their applicability in secure communications
Through an experimental sweep of the fractional order, we found that the minimum value of υ for which the system exhibits a chaotic behavior is observed for υ > υ0 ≈ 0.933
Summary
Chaotic discrete-time systems (maps) have received considerable attention over the last two decades due to their many applications in secure communications [1,2,3,4] and control [5]. Numerous maps have been proposed throughout the years including Hénon map [6], Lozi system [7], generalized Hénon map [8], Baier–Klein system [9], Stefanski map [10], Rössler map [11], and Wang map [12]. These maps exhibit a chaotic behavior in the sense that their trajectories are highly dependent on the system’s initial conditions. Since the topic of fractional discrete calculus is still new, to the best of our knowledge, very few fractional order chaotic maps have been proposed in the literature such as [18,19,20,21]
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