## 4 Applications in Optimization
### Differential Evolution (DE)
#### 4.2.4 Comparison of PSO with DE  
Having presented the PSO and DE optimization methods, it is natural to compare them.  
In [79], Senkerik et al. compared the swarm-based PSO technique to Differential Evolution on the task described above involving optimal evolutionary tuning of controller parameters for the stabilization of a Lozi discrete chaotic system. The novelty of this approach was that the most utilized examples of evolutionary/swarm based algorithms were compared directly on the highly nonlinear and complex multimodal optimization and simulation task. The simulations and comparisons were performed for three different required final behaviors of the chaotic system which were simulated in the following form:  
* Case study 1: p-1 UPO, Lozi map as controlled system with \(CF_{SIMPLE}\) (57)
* Case study 2: p-2 UPO with the same control
* Case study 3: higher order p-4 UPO with the same control Within their research, a total of fifty simulations for each experiments was performed in the _Wolfram Mathematica_ environment. All experiments used different initializations, i.e., a different initial population was generated in each run of DE/PSO.  
The parameter settings for DE were population size = 25, \(F=0.5\), \(Cr=0.9\), number of generations = 300, and CF Evaluations (CFE) = 7500.  
For PSO, they were population size = 25, \(c_{1}=c_{2}=2.0\), number of iterations = 300, and CF Evaluations (CFE) = 7500.  
All simulations were successful and provided new optimal settings for the ETDAS control method, securing the fast stabilization of the chaotic system at the required behaviors, which were p-1 UPO (stable state), p-2 UPO (oscillation between two values), and p-4 UPO.  
These results allowed the authors to compare DE and PSO in each case.  
* Case Study 1: from the statistical point of view, the performance of both heuristics was similar. Nevertheless, DE converged towards the optimal solution in only twenty generations, and all fifty runs provided identical results. The convergence for PSO was much slower. Therefore, DE is more suitable for online control of nonlinear chaotic dynamics.
* Case Study 2: as in Case Study 1, the structure asnd simulation outputs showed similar features, with an even larger difference in favor of the DE algorithm.
* Case Study 3: dealing with the most complex and highly nonlinear dynamics, this case showed several interesting features. The classical geometrical (vector crossover)-based evolutionary algorithm was stacked in many suboptimal solutions, with only a small chance of leaving this area of the solution space. The PSO (swarm-based) algorithm was statistically better in the searching process, with a lower standard deviation and range of CF. Even when DE found a lower final CF value, PSO seemed to be a better and more robust choice for optimization in extremely nonlinear solution spaces due to its better natural exploration ability.  
Senkerik et al. in [76] continued this comparison, using both \(CF_{SIMPLE}\) and \(CF_{UNIVERSAL}\):  
* Case Study 4: p-1 UPO, Lozi map as controlled system with \(CF_{UNIVERSAL}\) (58)
* Case Study 5: p-2 UPO with the same control
* Case Study 6: higher order p-4 UPO with the same control  
The authors found that \(CF_{UNIVERSAL}\) led to faster and more precise stabilization of the chaotic system.