## 4 Applications in Optimization
### Differential Evolution (DE)
#### 4.2.3 Evolutionary Chaos Control Optimization  
An original study introduced by Zelinka's team sought to utilize hidden chaotic dynamics in pseudo-random sequences generated via chaotic map to help the differential evolution algorithm search for the best controller settings for the same chaotic system. In (Zelinka, 2018; Zelinka, 2018; Zelinka, 2018), they considered (55) with the standard parameter values of \(a=1.7\) and \(b=0.5\), and tried to stabilize desired Unstable Periodic Orbits (UPO) using the chaos-driven DE for tuning the parameters of the Extended Delay Feedback (ETDAS) control method proposed by Pyragas (2018). \[\mathcal{L}_{a,b}\left( \begin{array}{c}X\\ Y\end{array} \right)=\left( \begin{array}{c}1-a\lvert X\rvert+Y,\\ bX.\end{array} \right) \tag{55}\] \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**DE Version** & **Avg. CF** & **Median CF** & **Std. Dev.** & **Max CF** & **Min CF** \\ \hline Canonical DE & 21,259.0 & 21,234.1 & 94.1 & 21,588.2 & 21,089.2 \\ Chaos DE Lozi & 21,107.6 & 21,107.4 & 25.4 & 21,160.7 & 21,059.0 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Average CF values for generation Nos. 50, 100, 150 and 200 from all 100 runs of Chaos DE and Canonical DE. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Generation No.:** & **50** & **100** & **150** & **200** \\ \hline Avg. CF for Canonical DE & 25,320.1 & 22,719.9 & 21,619.8 & 21,259.0 \\ Avg. CF for Chaos DE Lozi & 24,707.1 & 21,992.1 & 21,265.2 & 21,107.6 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Statistical results of all 100 runs of Chaos DE and Canonical DE. This control method takes the form  
\[\left\{ \begin{array}{l}x_{n+1}=1-a|X_{n}|+bY_{n}+F_{n},\\ F_{n}=K[(1-R)S_{n-m}-x_{n}],\\ S_{n}=x_{n}+RS_{n-m},\end{array} \right. \tag{56}\]  
where \(K\) and \(R\) are adjustable constants which have to be evolutionary tuned, \(F\) is the perturbation, \(S\) is provided by a delay equation utilizing the previous states of the system, and \(m\) is the period of the \(m\)-periodic orbit to be stabilized. The perturbation \(F_{n}\) in Equation (56) may have an arbitrarily large value, which can cause divergence of the system outside the output interval of the Lozi map system \([-1.4,1.4]\). Therefore, \(F_{n}\) should have a value between \([-F_{max},F_{max}]\). The suitable \(F_{max}\) value was obtained from the evolutionary optimization process. In these articles, two cost function designs, \(CF_{SIMPLE}\) and \(CF_{INIT}\), were utilized and compared. The goal of the first basic \(CF_{SIMPLE}\), which can be used only for the stabilization of p-1 orbit, was to minimize the area created by the difference between the required state and the real system output on the whole simulation interval \([0,\tau_{i}]\):  
\[CF_{SIMPLE}=\sum_{t=0}^{\tau_{i}}\lvert TS_{t}-AS_{t}\rvert, \tag{57}\]  
where \(TS_{t}\) is the target state and \(AS_{t}\) the actual state. Nevertheless, this simple approach has one prominent disadvantage, which is the inclusion of the initial chaotic transient behavior of the not-yet-stabilized system in the cost function value. To avoid this disadvantage, a universal cost function suppressing the initial chaotic transient behavior was introduced to ensure the stabilization of either the p-1 orbit (stable state) or the higher periodic orbit while retaining the possibility of adding penalization rules:  
\[CF_{UNIVERSAL}=pen_{1}+\sum_{t=\tau_{1}}^{\tau_{2}}\lvert TS_{t}-AS_{t}\rvert, \tag{58}\]  
where \(\tau_{1}\) is the first minimal value of the difference between TS and AS and \(\tau_{2}\) indicates the end of the optimization interval (\(\tau_{1}+\tau_{s}\)), with \(pen_{1}=0\) if \(\tau_{i}-\tau_{2}\geq\tau_{s}\) and \(pen_{1}=10*(\tau_{i}-\tau_{2})\) if \(\tau_{i}-\tau_{2}<\tau_{s}\) (i.e., late stabilization). After hundreds of numerical tests, the researchers on Zelinka's team concluded that embedding the chaotic dynamics into the differential evolution algorithm in the form of a chaotic pseudo-random number generator may help to improve their performance and robustness.