# The Fractional Form of the Tinkerbell Map Is Chaotic
###### Abstract  
This paper is concerned with a fractional Caputo-difference form of the well-known Tinkerbell chaotic map. The dynamics of the proposed map are investigated numerically through phase plots, bifurcation diagrams, and Lyapunov exponents considered from different perspectives. In addition, a stabilization controller is proposed, and the asymptotic convergence of the states is established by means of the stability theory of linear fractional discrete systems. Numerical results are employed to confirm the analytical findings.  
fractional discrete calculus; discrete chaos; Tinkerbell map; bifurcation; stabilization +
Footnote †: journal: Applied Sciences  
1  
[PERSON] and [PERSON] 2  
[PERSON] and [PERSON] 2018  
[PERSON] and [PERSON] 2  
[PERSON] and [PERSON]  
studied, and applied over the last five years are the fractional logistic map [12], the fractional Henon map [13], the generalized hyperchaotic Henon 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]. An application of a generalized fractional logistic map to data encryption and its FPGA implementation was achieved in [21].  
In this paper, we are interested in the Tinkerbell discrete-time chaotic system, which is of the form:  
\[\left\{\begin{array}{l}x\left(n+1\right)=x^{2}\left(n\right)-y^{2}\left(n \right)+\alpha x\left(n\right)+\beta y\left(n\right),\\ y\left(n+1\right)=2x\left(n\right)y\left(n\right)+\gamma x\left(n\right)+ \delta y\left(n\right),\end{array}\right. \tag{1}\]  
where \(\alpha\), \(\beta\), \(\gamma\), and \(\delta\) are system parameters and \(n\) represents the discrete iteration step. It is rumored that the map (1) derives its name from the famous Cinderella story, as the trajectory followed by the map resembles that of [PERSON] appearing in the movie adaptation of the fairy tale. The Tinkerbell map has been studied by many as it exhibits very rich dynamics including a chaotic behavior and a range of periodic states. For instance, its bifurcation subject to different scenarios and initial settings has been studied in [22; 23; 24; 25]. A more comprehensive study was performed in [26]. The authors identified conditions for the existence of fold bifurcation, flip bifurcation, and Hopf bifurcation in the Tinkerbell map.  
In order to visualize the dynamics of the map (1), we resort to phase plots, bifurcation diagrams, and Lyapunov exponent estimation. We assume parameter values \(\alpha=0.9\), \(\beta=-0.6013\), \(\gamma=2\), and \(\delta=0.5\) and initial states \(\left(x\left(0\right),y\left(0\right)\right)=\left(-0.72,-0.64\right)\). The results are depicted in Figure 1. The Tinkerbell map's phase plot is depicted in Figure 1a. Based on Figure 1b, we can see that the estimated Lyapunov exponents of (1) are given by \(\lambda_{1}\approx 0.2085\) and \(\lambda_{2}\approx-0.4925\). It is well known that a positive Lyapunov exponent indicates a chaotic behavior. The remaining parts of Figure 1 depict the bifurcation diagrams of the map (1) with respect to different parameters. These diagrams confirm that the map exhibits a range of different behaviors.  
It should be clear to the reader that the Tinkerbell map has rich dynamics and is heavily dependent on its parameters, as well as the initial setting. The main objective of this paper is to investigate the fractional Caputo-difference form of the Tinkerbell map in order to benefit from the added degrees of freedom due to the fractional nature. It is expected that the fractional form will have even richer dynamics and may consequently be more suitable for applications that require a higher entropy level such as data/image encryption.