## 7 Networks
### Solitary States
#### 7.3.2 Solitary State Chimeras  
Solitary states can additionally be found in the unexpected form of solitary state chimeras (SSC). Rybalova et al. [149] showed that this chimeric state can appear in a two-layer network of nonlocally coupled nonlinear chaotic maps. One of the coupled networks represents a ring of nonlocally coupled Lozi maps that individually shows a special spatiotemporal pattern, called the solitary state, when there is no coupling between the subnetworks. The authors analyzed the bifurcation mechanism of the solitary state appearance in the network of coupled Lozi maps, showing that this particular state is responsible for the formation of the solitary state chimera in the two-layer network. They first considered the single-layer network (116) with \(N=1000\). Their numerical studies showed that the solitary state's emergence is related to the structural changes in the chaotic attractor of the individual elements in this ensemble. In turn, these changes are caused by the impact of the \(2P\) nodes which the solitary state oscillator is coupled with. It can be argued that the solitary state regime is related to the formation of a new attractor in the phase space, which the network oscillators can reach (or not) depending on their initial conditions. Their numerical simulations revealed the mechanism of solitary state appearance in a ring of nonlocally coupled Lozi maps (116). It was shown that the nonlocal coupling induces the bistability mode in the individual nodes, at which two chaotic attractors with different basins of attraction co-exist in the phase space of this network. The first attractor corresponds to the Lozi attractor in the individual element without coupling. The second attractor emerges only in the presence of nonlocal coupling; thus, it promotes the appearance of solitary states. Randomly chosen initial conditions provide the possibility of the initial states of a group of oscillators belonging to the basin of attraction of the second attractor; in this way, solitary state regimes can be realized. Next, they considered a complex system consisting of three coupled rings of nonlocally coupled chaotic maps. This multilayer network is described by the following equations:  
\[\left\{ \begin{array}{l}x_{n+1}^{i}=f(x_{n}^{i},y_{n}^{i})+\frac{\sigma_{1}}{ 2P}\sum_{j=i-P}^{j=i+P}\Bigl{[}f(x_{n}^{j},y_{n}^{j})-f(x_{n}^{i},y_{n}^{i}) \Bigr{]}+\gamma_{1}F_{n}^{i},\\ y_{n+1}^{i}=bx_{n}^{i},\\ u_{n+1}^{i}=f(u_{n}^{i},v_{n}^{i})+\frac{\sigma_{2}}{2R}\sum_{j=i-R}^{j=i+R} \Bigl{[}f(u_{n}^{i},v_{n}^{i})-f(u_{n}^{i},v_{n}^{i})\Bigr{]}+\gamma_{2}G_{n}^{ i},\\ v_{n+1}^{i}=bx_{n}^{i},\\ z_{n+1}^{i}=f(z_{n}^{i},s_{n}^{i})+\frac{\sigma_{3}}{2P}\sum_{j=i-T}^{j=i+T} \Bigl{[}f(z_{n}^{j},s_{n}^{i})-f(z_{n}^{i},s_{n}^{i})\Bigr{]}+\gamma_{3}H_{n}^ {i},\\ s_{n+1}^{i}=bx_{n}^{i},\\ \end{array} \right. \tag{126}\]  
The first system of equations in (126) specifies a ring network of nonlocally coupled Henon maps  
\[f(x_{n},y_{n})=1-ax_{n}^{2}+y_{n}, \tag{127}\]  
with \(a=1.4\), \(b=0.3\), \(\sigma_{1}=0.72\), and \(P=320\). The second pair of equations corresponds to the ring of nonlocally coupled Lozi maps (with \(f\) defined by (117)) and was analyzed for \(a=1.4\), \(b=0.3\), \(\sigma_{2}=0.206\), and \(R=180\). The third pair of equations determines the ring of nonlocally coupled Henon maps with \(a=1.4\), \(b=0.3\), \(\sigma_{1}=0.295\), and \(T=320\). The first two rings are coupled inertially via the coupling functions \(F_{n}^{i}=-G_{n}^{i}=u_{n}^{i}-x_{n}^{i}\) with the coupling coefficients \(\gamma_{1}\) and \(\gamma_{2}\). The third ring node is connected unidirectionally to the first ring units by the coupling term \(\gamma_{3}H_{n}^{i}\), where  
\[H_{n}^{i}=f(x_{n}^{i},y_{n}^{i})-f(z_{n}^{i},s_{n}^{i}), \tag{128}\]  
defines the diffusive coupling with coupling coefficient \(\gamma_{3}\). The coupling parameters \(\sigma_{1,2,3}\) and \(P,R,T\) have the same meaning as in the description of system (116).