## 7 Networks
### Synchronization
#### 7.1.7 Synchronization of Globally Coupled Maps  
Roy and Acharyya [141] considered a network of Lozi maps periodically synchronized by the periodic variation of one of the parameters. Interestingly, it was observed that, due to the variation of the parameter with time, the transition from a desynchronized state to a synchronized state occurs at lower values of the global coupling strength.  
The system transition to the synchronized state in globally coupled Lozi maps under parametric excitation can provide a better way to achieve the synchronized state in the case of various natural and well known examples of synchronizing systems.  
The global coupling in the Lozi maps (3)  
\[\left\{\begin{array}{l}x_{t+1}=1-a(t)|x_{t}|+y_{t}\equiv f(x_{t},y_{t}),\\ y_{t+1}=bx_{t},\end{array}\right. \tag{113}\]  
is introduced by the term \(\epsilon h_{t}\), and is described by the equation  
\[\left\{\begin{array}{l}x_{t+1}^{i}=(1-\epsilon)f(x_{t}^{i},y_{t}^{i})+ \epsilon h_{t}\\ y_{t+1}^{i}=bx_{t}^{i},\end{array}\right. \tag{114}\]  
where  
\[h_{t}=\frac{1}{N}\sum_{j=1}^{N}f(x_{t}^{j},y_{t}^{j}). \tag{115}\]  
Here, the parameter \(\epsilon\) represents the strength of the global coupling of the maps, \(x_{t}^{i}\) and \(y_{t}^{i}\) are the values of the state variables at the \(i^{th}\) lattice point \(i=(1,2,\ldots,N)\) at discrete time \(t\), and the parameter \(\epsilon\) represents the strength of the global coupling of the maps.  
Synchronization in this system of equations at iterate \(n\) arises when  
\((x_{n}^{i},y_{n}^{i})=(x_{n}^{j},y_{n}^{j})\), \(\forall(i,j)\). Note that synchronization of the \(x\) variable implies synchronization of the \(y\) variable.  
Note that in a previous work Cano and Cosenza studied the behavior of a large number (\(N=1000\)) of globally coupled Lozi maps when the parameter \(a\) of the Lozi maps remains constant over time [142]. They observed four desynchronized states, chimeras (see Section 7.2), clusters, and synchronized states in increasing order of the coupling strength \(\epsilon\). Although the parameters \(a\) and \(b\) of the map were chosen such that the individual mapping was in the chaotic range, depending on the strength of the global coupling parameter \(\epsilon\) the behavior of the equilibrium state will be one of four states.  
Roy and Acharyya, instead, were only interested in observing how the synchronization of globally coupled Lozi maps changes when time dependence of the parameter is introduced.  
Time dependence of the parameter \(a(t)\) is introduced by considering periodically kicked \(a(t)\) of time period \(T\), i.e., \(a(t)\) varies with time as follows:  
\(a(t)=a_{2}\) always except at times, \(t=T,2T,3T,4T,\ldots\), etc. when \(a\) takes the value \(a_{2}\). Therefore, the periodically kicked \(a(t)\) of time period \((T)\) follows:  
\(\bullet\)\(a(t)=a_{2}\) at every \(t=nT\)  
\(\bullet\)\(a(t)=a_{1}\) otherwise  
where \(n\in\mathbb{Z}^{+}\), \(T\in\mathbb{Z}^{+}\).  
The values of \(a_{1}\) and \(a_{2}\) were chosen such that the map was in the chaotic region for \(a_{1}\) and in the fixed point region for \(a_{2}\). Thus, the parameter \(a\) of the Lozi map was forced to change its value from \(a_{1}=1.4\) to \(a_{2}=1.0\) after a regular interval of time.  
They found that the transition from a desynchronized to a synchronized state changes as the value of \(\epsilon\) crosses \(\epsilon_{c}\) from below, and that this change occurs at lower values of coupling strength (\(\epsilon_{c}\)) compared to the case of time-independent \(a\). They observed that the value of \(\epsilon_{c}\) increases as the time period \((T)\) of the periodically kicked \(a(t)\) increases and approaches the value that corresponds to the value of constant \(a_{1}=1.4\).