## 2 The Lozi Chaotic Map
### Fractional Lozi Map
#### 2.5.1 Classic Fractional Lozi Map  
The first published example of fractional map used a constant fractional order \(\
u\).  
Using the Caputo-like delta difference  
\[\begin{array}{rl}{}^{c}\Delta_{a}^{\
u}X(t)&=\Delta_{a}^{-(n-\
u)}\Delta^{n}X (t),\\ &=\frac{1}{\Gamma(n-\
u)}\sum_{s=a}^{l-(n-\
u)}(t-s-1)^{(n-\
u-s)}\Delta^{n}X( t),\end{array}\]  
Khennaoui et al. Khennaoui et al. (2018) defined the fractal Lozi map as follows:  
\[\left\{\begin{array}{rl}{}^{c}\Delta_{a}^{\
u}x(t)&=\alpha|x(t-1+\
u)|+y(t-1+ \
u)+1-x(t-1+\
u),\\ {}^{c}\Delta_{a}^{\
u}y(t)&=\beta x(t-1+\
u)-y(t-1+\
u),\end{array}\right. \tag{11}\]  
for \(0<\
u\leq 1\) and \(t\in\mathbb{N}_{a+1-\
u}=\{a+1-\
u,a+2-\
u,\ldots\}\). In this example the fractional order of the two fractional differences is identical. This system is commonly referred to as a  
Figure 7: Quasi-periodic and chaotic orbits of the area-preserving map (3) in the plane \((x,y)\) when \(a=1.1\) and \(b=-1\). Initial values: green (most extended chaotic orbit): \(x_{0}=2.1\), \(y_{0}=2.0\); red (four-ellipse quasiperiodic orbit inside the green orbit): \(x_{0}=2.5\), \(y_{0}=1.0\); blue (32-ellipse quasiperiodic orbit outside the green orbit): \(x_{0}=0.87\), \(y_{0}=4.36\); magenta (chaotic orbit ): \(x_{0}=0.78\), \(y_{0}=5.05\); red (nine-ellipse quasiperiodic orbit inside the magenta chaotic orbit): \(x_{0}=1.2\), \(y_{0}=6.5\); light blue (nine-ellipse quasiperiodic orbit inside the nine red ellipses): \(x_{0}=2.0\), \(y_{0}=6.0\).  
commensurate system. The following equivalent discrete integral equation of such a map can be obtained:  
\[\left\{\begin{array}{rl}&x(t)=x(a)+\frac{1}{\Gamma(v)}\sum_{j=1}^{t-v}(t-s-1)^{ (v-1)}[-\alpha]x(t-1+v)|+y(t-1+v)+1-x(t-1+v)],\\ &y(t)=y(a)+\frac{1}{\Gamma(v)}\sum_{s=0-v}^{t-v}(t-s-1)^{(v-1)}[\beta x(t-1+v )-y(t-1+v)],\end{array}\right. \tag{12}\]  
where \(\frac{(t-s-1)^{(v-1)}}{\Gamma(v)}\) is the discrete kernel function  
\[\frac{(t-s-1)^{(v-1)}}{\Gamma(v)}=\frac{\Gamma(t-s)}{\Gamma(v)\Gamma(t-s-v+1)}\]  
and \(a=0\) yields the numerical formula  
\[\left\{\begin{array}{rl}&x(n)=x(0)+\frac{1}{\Gamma(v)}\sum_{j=1}^{n}\frac{ \Gamma(n-j+v)}{\Gamma(n-j+1)}[-\alpha]x(j-1)|+y(j-1)+1-x(j-1)],\\ &y(n)=y(0)+\frac{1}{\Gamma(v)}\sum_{j=1}^{n}\frac{\Gamma(n-j+v)}{\Gamma(n-j+1 )}[\beta x(j-1)-y(j-1)].\end{array}\right. \tag{13}\]  
Extended numerical simulations of this fractional map have shown that the value of the fractional order \(v\) modifies the general shape of the bifurcation diagram (compared to the non-fractional map) as well as the length of the chaotic interval. If \(v=0.98\), the bifurcation diagram appears similar the corresponding integer diagram, with the exception of a small widening in the interval where chaos is observed. For smaller values of \(v\), when \(0\leq\alpha\leq 0.5\) the orbit no longer goes to a fixed point, unlike the non-fractional map. In fact, as \(n\) increases it can be observed that the trajectory becomes unbounded. A major difference between the bifurcation diagram of the integer and fractional maps concerns the interval over which chaos is observed, which becomes slightly smaller as \(v\) decreases.  
Ibrahim and Baleanu [40] used linear interpolation of quadratic and absolute value functions to define and study a combined Henon-Lozi fractional map.