# The Fractional Form of the Tinkerbell Map Is Chaotic
## 4 Control of the Fractional Tinkerbell Map  
In this section, we show that the proposed fractional Tinkerbell can be stabilized by means of a simple adaptive feedback controller. In order to be able to establish the asymptotic convergence of the controlled states towards zero, we first need to recall some important results from the literature concerning the asymptotic stability of fractional discrete systems. Since fractional discrete calculus is still relatively new, the existing literature related to stability is very limited. There are two main ways of establishing asymptotic stability. The first relies on the linearity of the system and places conditions on the eigenvalues of the Jacobian [31]. The second scheme is a generalization of the well-known Lyapunov direct method [32]. Although, the Lyapunov method is powerful and can support different types of systems, its has yet to be established for delayed fractional discrete systems, which renders it unusable for the system at hand. Hence, our objective here is to design the control laws to linearize the system, which will allow us to use the stability theory of linear systems. The following theorem summarizes the result of [31]. **Theorem 1**. : _The zero equilibrium of the linear fractional discrete system:_  
\[{}^{C}\Delta_{a}^{v}F\left(t\right)=MF\left(t+v-1\right), \tag{9}\]  
_where \(F(t)=\left(f_{1}(t),...,f_{n}(t)\right)^{\mathsf{T}}\), \(0<v\leq 1\), and \(M\in\mathbb{R}^{n\times n}\), is asymptotically stable if the eigenvalues \(\lambda\) of \(M\) satisfy:_  
\[\lambda\in\left\{z\in\mathbb{C}:\ |z|<\left(2\cos\frac{|\arg z|-\pi}{2-v} \right)^{v}\text{ and }\ |\arg z|>\frac{v\pi}{2}\right\} \tag{10}\]  
_for all \(t\in\mathbb{N}_{a+1-v}\)._  
Consider the controlled version of (7) given by:  
\[\left\{ \begin{array}{l}{}^{C}\Delta_{a}^{v}x\left(t\right)=x^{2}\left(t-1+v \right)-y^{2}\left(t-1+v\right)+\left(\alpha-1\right)x\left(t-1+v\right)\\ \qquad\qquad\qquad+\beta y\left(t-1+v\right)+u_{x}\left(t-1+v\right),\\ {}^{C}\Delta_{a}^{v}y\left(t\right)=2x\left(t-1+v\right)y\left(t-1+v\right)+ \gamma x\left(t-1+v\right)\\ \qquad\qquad\qquad+\left(\delta-1\right)y\left(t-1+v\right)+u_{y}\left(t-1+v \right),\end{array} \right. \tag{11}\]  
Figure 7: The largest Lyapunov exponent as a function of the fractional order \(v\) for the same parameters and initial conditions in Figure 6.  
where \(u_{x}\left(t\right)\) and \(u_{y}\left(t\right)\) are adaptive control terms. The following theorem presents the proposed control laws. **Theorem 2**. : _The states of the controlled 2D fractional Tinkerbell map (11) are guaranteed to converge towards zero asymptotically subject to:_  
\[\left\{ \begin{array}{l}u_{x}\left(t\right)=y^{2}\left(t\right)-x^{2}\left(t \right),\\ u_{y}\left(t\right)=-2x\left(t\right)y\left(t\right)-\gamma x\left(t\right). \end{array} \right. \tag{12}\]  
Proof. : Substituting (12) into (11) yields the dynamics:  
\[\left\{ \begin{array}{l}{}^{C}\Delta_{a}^{v}x\left(t\right)=\left(\alpha-1 \right)x\left(t-1+v\right)+\beta y\left(t-1+v\right),\\ {}^{C}\Delta_{a}^{v}y\left(t\right)=\left(\delta-1\right)y\left(t-1+v\right), \end{array} \right. \tag{13}\]  
or more compactly:  
\[{}^{C}\Delta_{a}^{v}\left(x\left(t\right),y\left(t\right)\right)^{T}=A\left( x\left(t\right),y\left(t\right)\right)^{T}, \tag{14}\]  
with:  
\[A=\left( \begin{array}{cc}\alpha-1&\beta\\ 0&\delta-1\end{array} \right). \tag{15}\]  
The eigenvalues of Aare simply \(\lambda_{1}=\alpha-1\) and \(\lambda_{2}=\delta-1\). It is straight forward to see that these eigenvalues satisfy the conditions of Theorem 1. Consequently, the zero solution of (13) is asymptotically stable, and the states of the controlled map (11) are asymptotically stabilized. The result of Theorem 2 can be easily put to the test. Consider, for instance, parameters \(\left(\alpha,\beta,\gamma,\delta\right)=\left(0.9,-0.6013,2,0.5\right)\), initial conditions \(\left(x\left(0\right),y\left(0\right)\right)=\left(-0.72,-0.64\right)\), and fractional order \(v=0.98\).