## 7 Networks
### Solitary States
#### 7.3.1 Existence of Solitary States  
Semenova et al. [147], revisiting their previous work, studied (116). They showed that the ensemble of nonlocally coupled Lozi maps demonstrates the solitary state for specific values of the coupling parameters. Coupling changes the properties of partial elements and leads to the bistability, though the Lozi map does not have this property in its uncoupled form. The emergence of solitary states is accompanied by a second attracting set arising for the ensemble element. Using a ring network of nonlocally coupled Lozi maps as an example, Rybalova and Strelkova [148] explored the robustness of solitary states against heterogeneity in the parameters of the local dynamics or coupling strength. They found that if these network parameters are continuously modulated by noise, then solitary states are suppressed as the noise intensity increases. However, these states may persist in the case of static randomly distributed system parameters for a wide range of the distribution width. Domains of solitary state existence were constructed in the parameter plane of coupling strength and noise intensity using a cross-correlation coefficient. They numerically studied the spatiotemporal dynamics of a one-dimensional ensemble (a ring network) of nonlocally coupled discrete-time systems. This network is described by the following system of equations, a generalization of (116):  
\[\left\{ \begin{array}{l}x_{i}^{n+1}=F(x_{i}^{n},y_{i}^{n})+\frac{c_{i}^{n}}{2R} \sum_{j=i-P}^{j=i+P}\Bigl{[}F(x_{j}^{n},y_{j}^{n})-F(x_{i}^{n},y_{i}^{n})) \Bigr{]},\\ y_{i}^{n+1}=G(x_{i}^{n},y_{i}^{n}),\end{array} \right. \tag{122}\]  
where \(x_{i}^{n}\), \(y_{i}^{n}\) are dynamical variables, \(i=1,2,\ldots,N\) denotes the element number, \(N=1000\) is the total number of elements in the ensemble, and \(n\) is the discrete time. The local dynamics of every element \(i\) in the ring (122) are described by the two-dimensional Lozi map  
\[\left\{ \begin{array}{l}F(x_{i}^{n},y_{i}^{n})=1-\alpha_{i}^{n}|x_{i}^{n}|+y_ {i}^{n},\\ G(x_{i}^{n},y_{i}^{n})=\beta x_{i}^{n},\end{array} \right. \tag{123}\]  
where \(\alpha_{i}^{n}>0\) and \(\beta>0\) are the control parameters. The Lozi map demonstrates a quasihyperbolic chaotic attractor which follows directly from a border-collision bifurcation developed from a stable four-periodic orbit [10; 12] and exists within the parameter range \(\alpha\in[1.3,1.8]\). In their simulations, Rybalova and Strelkova fixed \(\beta=0.3\) for all the network elements. The heterogeneities in the parameters \(\alpha_{i}^{n}\) and \(\sigma_{i}^{n}\) were realized using two independent noise sources with a uniform distribution and a standard normal (Gaussian) distribution, while noise-modulated parameters were determined by the following expressions:  
\[\alpha_{i}^{n}=\alpha_{0}+D_{\alpha}\xi_{i}^{n}, \tag{124}\]  
\[\sigma_{i}^{n}=\sigma_{0}+D_{\alpha}\xi_{i}^{n}, \tag{125}\]  
where \(\alpha_{0}=1.4\) is the mean value of the local dynamics parameter, \(\sigma_{0}\) is the mean value of the coupling strength, which was varied in the numerical experiments, \(\xi_{i}^{n}\) is a noise generator with a uniform distribution in the interval \([-1,1]\), and \(D\) is the noise intensity (i.e., the width of the interval within which random source values are distributed). Calculations were performed for fifty different realizations of the initial conditions \(x_{i}^{0}\) and \(y_{i}^{0}\) randomly distributed in the interval \([-0.5,0.5]\). For each set of initial conditions, different uniform noise realizations were used for parameter modulation or to realize an inhomogeneous distribution of the system parameters. This allowed satisfactory statistical data to be obtained. They considered many different situations, including noise-modulated local dynamics parameters, noise-modulated coupling strengths, the case of randomly distributed local dynamics parameters, the case of randomly distributed coupling strengths, and localized inhomogeneity of the network parameters.