# The Fractional Form of the Tinkerbell Map Is Chaotic
## 3 Dynamics of the Fractional Tinkerbell Map  
In this section, we will employ numerical tools to assess the dynamics of the proposed fractional Tinkerbell map (7). For that, we will need a discrete numerical formula that allows us to evaluate the states of the map in fractional discrete time. According to [29] and other similar studies, we can evaluate (7) numerically as:  
\[\left\{ \begin{array}{l}x\left(n\right)=x\left(0\right)+\frac{1}{\Gamma\left( v\right)}\sum\limits_{j=1}^{n}\frac{\Gamma\left(n-j+v\right)}{\Gamma\left(n-j+1 \right)}\left[x^{2}\left(j-1\right)-y^{2}\left(j-1\right)\right.\\ \qquad\qquad+\left(\alpha-1\right)x\left(j-1\right)+\beta y\left(j-1\right) \right],\\ y\left(n\right)=y\left(0\right)+\frac{1}{\Gamma\left(v\right)}\sum\limits_{j=1 }^{n}\frac{\Gamma\left(n-j+v\right)}{\Gamma\left(n-j+1\right)}\left[2x\left(j -1\right)y\left(j-1\right)\right.\\ \qquad\qquad\left.+\gamma x\left(j-1\right)+\left(\delta-1\right)y\left(j-1 \right)\right],\end{array} \right. \tag{8}\]  
where we assumed \(a=0\) for simplicity. This yields an initial-value problem similar to that of [30], which allows us to use a similar discrete integral equation. Using Formula (8), we may obtain the states of the fractional Tinkerbell map and consequently produce time series plots of the states, phase-space plots, and bifurcation diagrams. We start with a simple case where the parameters and initial conditions are identical to those adopted in the standard case, i.e., \(\left(\alpha,\beta,\gamma,\delta\right)=\left(0.9,-0.6013,2,0.5\right)\) and \(\left(x\left(0\right),y\left(0\right)\right)=\left(-0.72,-0.64\right)\). Given the fractional  
Although the phase plots give an indication of the behavior of the map, it is not until we visualize the bifurcation of the map subject to different parameters that a more complete picture forms. We choose the parameter \(\beta\) as the critical parameter and varied it over the range \(\beta\in[-0.6,-0.1]\) in steps of \(\Delta\beta=0.0025\). The process may be easily repeated for other parameters. The bifurcation diagrams obtained using the same parameter and initial condition values from earlier are depicted in Figure 4. We observe that although the general dynamics remain similar, the intervals seem to become shorter as the fractional order is decreased. Even though these bifurcation diagrams suggest the existence of chaos in the fractional Tinkerbell map, they are not definitive. Generally, in order to prove the existence of chaos, we must use multiple tools including time series, phase portraits, Poincare maps, power spectra, bifurcation diagrams, Lyapunov exponents, etc. The next tool at our disposal is Lyapunov exponents. We calculate these exponents by means of the Jacobian method. It is well known that when \(\lambda_{max}\) is positive and the points in the corresponding bifurcation diagram are dense, the map is highly likely to be chaotic. Figure 5 shows the largest Lyapunov exponents corresponding to the same bifurcation diagrams depicted in Figure 4 in the \(x\)-\(\beta\) plane. We can observe clearly that for certain ranges of the parameter \(\beta\), chaos exists. Figure 4: Bifurcation diagrams of the fractional Tinkerbell map (7) with \(\beta\in[-0.6,-0.1]\) being changed in steps of \(\Delta\beta=0.0025\), parameters \((\alpha,\gamma,\delta)=(0.9,2,0.5)\), initial conditions \((x\left(0\right),y\left(0\right))=(-0.72,-0.64)\), and different fractional orders. Another interesting aspect is the effect of the fractional order on the dynamics of the map for a specific set of parameter values. We fix the parameters and initial conditions at \(\left(\alpha,\beta,\gamma,\delta\right)=\left(0.9,-0.6013,2,0.5\right)\) and \(\left(x\left(0\right),y\left(0\right)\right)=\left(-0.72,-0.64\right)\), respectively. Figure 6 shows the bifurcation plot with the critical parameter \(v\in\left[0,1\right]\) being changed in steps of \(\Delta v=0.005\).