Abstract
In this paper, we investigate the dynamics of a fractional order chaotic map corresponding to a recently developed standard map that exhibits a chaotic behavior with no fixed point. This is the first study to explore a fractional chaotic map without a fixed point. In our investigation, we use phase plots and bifurcation diagrams to examine the dynamics of the fractional map and assess the effect of varying the fractional order. We also use the approximate entropy measure to quantify the level of chaos in the fractional map. In addition, we propose a one-dimensional stabilization controller and establish its asymptotic convergence by means of the linearization method.
Highlights
Over the last few decades and since the Hénon map was first proposed [1], discrete-time chaotic dynamical systems have received a great deal of attention from numerous disciplines due to their ability to model various natural phenomena [2]
We show that the fractional order has a major impact on the chaotic range and the shape of hidden attractors
We have examined a fractional chaotic map based on the standard generic map proposed in [14], which exhibits rich dynamics and hidden chaotic attractors under different circumstances, i.e., with no fixed points, with a single fixed point or with two fixed points
Summary
Over the last few decades and since the Hénon map was first proposed [1], discrete-time chaotic dynamical systems have received a great deal of attention from numerous disciplines due to their ability to model various natural phenomena [2]. An interesting investigation was carried out in [14] on a rather general 2D map that can, under certain parameters, have no fixed points and possess hidden dynamics. Note that the term hidden attractors first came about in the investigation of continuous chaotic systems [15,16,17]. With the growing advancement in the field of discrete fractional calculus, a few studies have emerged considering the dynamics, control and applications of fractional chaotic maps [26,27,28,29,30,31,32]. As far as we aware, this is the first time that a fractional-order chaotic map without fixed points has been investigated. Throughout our analysis, we make use of numerical methods to confirm the findings
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