## 2 The Lozi Chaotic Map
### Advanced Dynamical Properties of the Dissipative Map (\(\,\mid b\mid\,<1\))  
An open region \(\mathcal{B}\) in the parameter space for which the Lozi map exhibits periodic points of least period \(n\) was rigorously determined for all \(n>13\) by Boronski, Kucharski and Ou [16] as follows:  
\[\mathcal{P}_{str}=\{(a,b)|\ b>0,b<l_{str}(a)\ \text{when}\ a<a_{0},\ \text{and}\ b<l_{tan}(a)\ \text{when}\ a>a_{0}\,\} \tag{7}\]  
and  
\[\mathcal{P}_{3}(a,b)=\left\{(a,b)|\,0\leq b\leq 1\,,\,a>l_{hyp}(b),\,\text{and}\,\,a>l _{3}(b)\right\} \tag{8}\]  
where \(a_{0}=\frac{2}{9}\left(2+3\sqrt{2}\right)\approx 1.78,l_{str}(a)=-2+\sqrt{2}a,l_{ tan}(a)=\frac{1}{8}\left(8a-3a^{2}-\sqrt{9a^{4}-16a^{3}}\right)\), \(l_{hyp}(b)=b+1\), and \(l_{3}(b)=\frac{1}{2}\left[1-b+\sqrt{(1-b)^{2}+4(1-b+b^{2})}\right]\). **Theorem 1** ([16]). : _For all \((a,b)\in\mathcal{B}=\mathcal{P}_{str}\cap\mathcal{P}_{3}\) the Lozi map \(\mathcal{L}_{a,b}\) has a periodic point of least period \(n\) for \(n=1,2,3\) and all \(n>13\)._  
A survey of all the papers describing the other dynamical or statistical properties of the Lozi map cannot be provided in this article because they are too numerous. Readers may refer to [1] for a compendium of results published between 1997 to 2013. Below, only a selection of particular and more recent results are pointed out. Several new results on bifurcations have been recently published since 2013. Botella-Soler et al. [17] showed that this map presents what they call _bisecting bifurcations_, that is, those which are mediated by an infinite set of neutrally stable periodic orbits (see Figures 5 and 6). Sushko et al. [18] investigated the bifurcation structure of the parameter plane in the vicinity of the curve related to a center bifurcation of the fixed point. A distinguishing property of the Lozi map is that it is conservative (see Section 2.4) at the parameter value  
Figure 5: Bifurcation diagram for b = 0.1. Notice that the bisecting bifurcations take place at \(a=1\pm b\). Figure 6: Bifurcation diagram for a = 1.5. Notice that the bisecting bifurcations take place at \(b=\pm(a-1)\). corresponding to this bifurcation. As a result, the bifurcation structure close to the center bifurcation curve is quite complicated. In particular, an attracting fixed point (focus) can coexist with various attracting cycles as well as with chaotic attractors, and the number of coexisting attractors increases as the parameter point approaches the center bifurcation curve. Their study contributed to the border collision bifurcation theory as well, as the Lozi map is a particular case of the 2D border collision normal form (2D-BCNF). The following 2D-BCNF family of difference equations \((x,y)\longrightarrow h(x,y)\) was presented by Glendinning and Simpson [19] as a canonical example:  
\[h(x,y)=\left\{ \begin{array}{l}F_{L}\left[\begin{array}{c}x\\ y\end{array}\right]+\left[\begin{array}{c}\mu\\ 0\end{array}\right],\quad x\leq 0,\\ \\ F_{R}\left[\begin{array}{c}x\\ y\end{array}\right]+\left[\begin{array}{c}\mu\\ 0\end{array}\right],\quad x\geq 0,\end{array} \right. \tag{9}\]  
where the parameter \(\mu\) controls the border-collision bifurcation and with  
\[F_{L}=\ \ \left[ \begin{array}{cc}\pi_{L}&1\\ -\
u_{L}&0\end{array} \right],\quad F_{R}=\left[ \begin{array}{cc}\pi_{R}&1\\ -\
u_{R}&0\end{array} \right].\]  
In their paper, they restricted their attention to the parameter values \(\pi_{R}\in\mathbb{R},\quad\pi_{L}>0\), \(\
u_{L}>0,\quad\
u_{R}>0,\quad\mu=1,\) for which \(f\) is invertible and orientation-preserving. The role of \(\mu\) is to control the border-collision bifurcation. In view of a linear rescaling, it is only necessary to consider it for the values \(\{-1,0,1\}\) (here, it is 1). The condition \(\pi_{L}>0\) is needed for the definition of the induced map. If \(\pi_{L}=-\pi_{R}\) and \(\
u_{L}=\
u_{L}\), then the 2D BCNF reduces to the Lozi map.