# Search Patterns Based on Trajectories Extracted from the Response of Second-Order Systems
## 7 Experimental Results
### Multimodal Functions
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline  & & **ABC** & **DE** & **CMAES** & **CSA** & **PSO** & **MFO** & **SOA** \\ \hline \multirow{3}{*}{\(f_{1}(\mathbf{x})\)} & **AB** & 8.9132622 & 0.7932535 & 2.8976 \(\times\) 10\({}^{-19}\) & 55.918504 & 0.2012803 & 27.983271 & 0.1119774 \\  & **MD** & 8.4392750 & 0.7993166 & 2.4779 \(\times\) 10\({}^{-19}\) & 57.038263 & 4.1459 \(\times\) 10\({}^{-23}\) & 25.365199 & 1.0714 \(\times\) 10\({}^{-10}\) \\  & **SD** & 2.6748059 & 0.1378538 & 1.5343 \(\times\) 10\({}^{-19}\) & 6.2032578 & 1.1022968 & 12.283254 & 0.2272439 \\ \hline \multirow{3}{*}{\(f_{2}(\mathbf{x})\)} & **AB** & 2 & 2 & 2 & 1,897,783.3 & 27.4 & 2 & 2 \\  & **MD** & 2 & 2 & 2 & 35,691.155 & 2 & 2 & 2 \\  & **SD** & \(9.9512\times 10^{-12}\) & 0 & 0 & 9,636,632.1 & 113.43495 & 0 & 0 \\ \hline \multirow{3}{*}{\(f_{3}(\mathbf{x})\)} & **AB** & 2 & 2 & 2 & 3,620,834.4 & 34.723128 & 2 & 2 \\  & **MD** & 2 & 2 & 2 & 676,981.46 & 9 & 2 & 2 \\  & **SD** & 2.3308 \(\times\) 10\({}^{-11}\) & 0 & 0 & 7,265,352.7 & 113.36672 & 0 & 0 \\ \hline \multirow{3}{*}{\(f_{4}(\mathbf{x})\)} & **AB** & 0.1371551 & 0.002 & 1.7942 \(\times\) 10\({}^{-6}\) & 0.0862919 & 2.2285 \(\times\) 10\({}^{-8}\) & 5,5194 \(\times\) 10\({}^{-10}\) & 1.164 \(\times\) 10\({}^{-11}\) \\  & **MD** & 0.1349076 & 0.01 & 0 & 0.0892256 & 0 & 0 & \(7.7118\times\) 10\({}^{-12}\) \\  & **SD** & 0.0399861 & 0.123 & 5.4833 \(\times\) 10\({}^{-6}\) & 0.0213332 & 1.2206 \(\times\) 10\({}^{-7}\) & 3,0231 \(\times\) 10\({}^{-9}\) & 1.0995 \(\times\) 10\({}^{-11}\) \\ \hline \multirow{3}{*}{\(f_{5}(\mathbf{x})\)} & **AB** & 13,781,291 & 1,331,987.7 & 22,307.195 & 44,274,761 & 82.539625 & 85.756149 & 71.964984 \\  & **MD** & 13,876,263 & 1,365,502.1 & 72,377516 & 46,153,728 & 81.698488 & 86.665615 & 72.362277 \\  & **SD** & 3,237,147.7 & 306,385.34 & 50,923,637 & 10,118,180 & 7.916726 & 3.2857088 & 0.9929591 \\ \hline \multirow{3}{*}{\(f_{6}(\mathbf{x})\)} & **AB** & 1.152 \(\times\) 10\({}^{85}\) & 5.850 \(\times\) 10\({}^{81}\) & 1.812 \(\times\) 10\({}^{83}\) & 6.429 \(\times\) 10\({}^{83}\) & 1.397 \(\times\) 10\({}^{81}\) & 3,0883 \(\times\) 10\({}^{81}\) & 1.0051 \(\times\) 10\({}^{81}\) \\  & **MD** & 4.622 \(\times\) 10\({}^{84}\) & 3.405 \(\times\) 10\({}^{81}\) & 7.928 \(\times\) 10\({}^{82}\) & 2.939 \(\times\) 10\({}^{8}\) & 5.977 \(\times\) 10\({}^{80}\) & 7.9607 \(\times\) 10\({}^{80}\) & 4.901 \(\times\) 10\({}^{80}\) \\  & **SD** & 1.685 \(\times\) 10\({}^{85}\) & 7.33 \(\times\) 10\({}^{81}\) & 3.047 \(\times\) 10\({}^{83}\) & 7.551 \(\times\) 10\({}^{83}\) & 2.197 \(\times\) 10\({}^{81}\) & 5.6651 \(\times\) 10\({}^{81}\) & 1.9457 \(\times\) 10\({}^{81}\) \\ \hline \multirow{3}{*}{\(f_{7}(\mathbf{x})\)} & **AB** & 30.033333 & 30 & 30 & 58.766666 & 30 & 33.6333333 & 30 \\  & **MD** & 30 & 30 & 30 & 30 & 59 & 30 & 30 & 30 \\  & **SD** & 0.18257419 & 0 & 0 & 1.77498583 & 0 & 5.4550409 & 0 \\ \hline \multirow{3}{*}{\(f_{8}(\mathbf{x})\)} & **AB** & 9.2 & 8.0666666 & 1.0666666 & 19,543.266 & 0.0333333 & 2000.03333 & 0 \\  & **MD** & 9 & 18 & 0 & 19,797 & 0 & 0 & 0 \\ \cline{1-1}  & **SD** & 2.5784250 & 1.9464084 & 3.1941037 & 2077.7459 & 0.1825741 & 484.23277 & 0 \\ \hline \multirow{3}{*}{\(f_{9}(\mathbf{x})\)} & **AB** & \(-745,05202\) & \(-1125,4815\) & \(-1127,8626\) & \(-725,09353\) & \(-1071,7869\) & \(-1031,2617\) & \(-1146,3478\) \\ \cline{1-1}  & **MD** & \(-743,4462\) & \(-1174,9722\) & \(-1132,5748\) & \(-719,10777\) & \(-1068,9596\) & \(-1033,6178\) & \(-1145,2467\) \\ \cline{1-1}  & **SDAccording to Table 1, the proposed SOA scheme obtain a better performance than ABC, CMAES, CSA, DE, MFO and PSO in functions \(f_{1}(\mathbf{x})\), \(f_{4}(\mathbf{x})\), \(f_{5}(\mathbf{x})\), \(f_{6}(\mathbf{x})\), \(f_{8}(\mathbf{x})\), \(f_{9}(\mathbf{x})\), \(f_{10}(\mathbf{x})\), \(f_{11}(\mathbf{x})\) and \(f_{12}(\mathbf{x})\). Nevertheless, the results of SOA exhibit similar as the obtained by DE, CMAES and MFO in functions \(f_{2}(\mathbf{x})\), \(f_{3}(\mathbf{x})\) and \(f_{7}(\mathbf{x})\).