## 7 Networks
### Chimeras  
In addition to the general synchronization presented above, the Lozi map can be used to construct networks of chaotic attractors, either alone or with the Henon map, which present unusual behaviors such as chimera and solitary states. In networks of chaotic attractors, chimeras consist of the coexistence of one or more clusters and a subset of desynchronized elements. In 2020, Parastesh et al. wrote a survey on chimeras [143]. In this section, we include more recent results. It is important to note that chimeras are not easy to find. In 2015 and 2016, Semenova et al. [144; 145] studied the following network of Lozi maps:  
\[\left\{ \begin{array}{l}x_{n+1}^{i}=f(x_{n}^{i},y_{n}^{i})+\frac{\sigma}{2P }\sum_{j=i-P}^{j=i+P}\Bigl{[}f(x_{n}^{j},y_{n}^{j})-f(x_{n}^{i},y_{n}^{i}) \Bigr{]},\quad i=1,2,\ldots,N,\\ y_{n+1}^{i}=bx_{n}^{i},\end{array} \right. \tag{116}\]  
with  
\[f(x_{n},y_{n})=1-a|x_{n}|+y_{n}, \tag{117}\]  
where \(N\) is the number of elements in the ensemble of coupled equations. The nonlocal coupling is characterized by the coupling strength \(\sigma\), number of neighbours \(2P\) (\(P\) neighbors on the either side of the \(i\)th element), and coupling range \(r=P/N\). They hypothesized that chimeric states can be obtained only in networks of chaotic non-hyperbolic systems, and are not found in networks of hyperbolic systems such as (116). This hypothesis was supported by their numerical simulations. However, they revisited this hypothesis later and found it to be wrong due to insufficient previous simulations (see Section 7.3.1). As mentioned above, in 2017 Cano and Cosenza [142] considered the autonomous system of globally coupled Lozi maps described by Equations (113) and (114), with \(a(t)\) remaining constant \(a\):  
\[\left\{ \begin{array}{l}x_{n+1}^{i}=(1-\epsilon)f(x_{n}^{i},y_{n}^{i})+ \epsilon h_{n},\\ y_{n+1}^{i}=bx_{n}^{i},\end{array} \right. \tag{118}\]  
with \(f\) defined by (117) and  
\[h_{n}=\frac{1}{N}\sum_{j=1}^{N}f(x_{n}^{i},y_{n}^{i}). \tag{119}\]  
In addition to synchronization, they observed the following collective states in this globally coupled system:  
1. _Clustering_: a dynamical cluster is defined as a subset of elements that are synchronized among themselves. In a clustered state, the elements in the system segregate into \(K\) distinct subsets that evolve in time, i.e., \(x_{n}^{i}=x_{n}^{j}=X_{n}^{\
u}\), \(\forall(i,j)\) in the \(v\)th cluster with \(\
u=1,\ldots,K\). 2. A _chimeric state_ consists of the coexistence of one or more clusters and a subset of desynchronized elements. 3. A _desynchronized or incoherent state_ occurs when \(x_{n}^{i}\
eq x_{n}^{j},\forall(i,j)\). They considered the system of nonlocally coupled Lozi maps described by  
\[\left\{ \begin{array}{l}x_{n+1}^{i}=(1-\epsilon)f(x_{n}^{i},y_{n}^{i})+\epsilon h _{n}^{i},\\ y_{n+1}^{i}=bx_{n}^{i},\end{array} \right. \tag{120}\]  
with  
\[h_{n}^{i}=\frac{1}{2k}\!\!\sum_{j=i-k}^{j=i+k}\!\!\left[f(x_{n}^{i},y_{n}^{i})- f(x_{n}^{i},y_{n}^{i})\right]\!, \tag{121}\]  
where the elements \(i=1,\ldots,N\) are located on a ring with periodic boundary conditions, \(\epsilon\) is the coupling parameter, \(k\) is the number of neighbors coupled on either side of site \(i\), and \(h_{n}^{i}\) is the local field acting on element \(i\). The presence of chimeric states in globally coupled networks of identical oscillators at first seemed counterintuitive because of the perfect symmetry of such a system. However, such networks are among the simplest extended systems that can exhibit chimeric behavior. Cano and Cosenza identified that the presence of global interactions can indeed allow for the emergence of chimera states in networks of coupled elements possessing chaotic hyperbolic attractors (such as Lozi maps) in cases where such states do not form with local interactions. Both chimeras and clusters can be interpreted as manifestations of the multistability of the resulting drive-response dynamics at the local level in systems with global interactions.