本文系统的研究了基于混沌映射的粒子群算法优化及其应用。论文首先对PSO算法的基本原理进行了全面而深入的剖析，详细阐述了其优化机制、优缺点以及在处理复杂优化问题时所面临的局限性。这些局限性主要包括易于陷入局部最优解、收敛速度较慢以及对于某些特定问题的求解能力有限，导致求解精度较低等。为了克服这些缺陷，论文进一步引入了混沌映射理论，深入探讨了多种混沌映射的非线性、遍历性和随机性等特性，并分析了这些混沌映射方法在优化算法设计中的潜在应用价值。

基于包括Tent混沌映射和Logistic混沌映射在内的多种混沌映射的特性，本文提出了一类新型的粒子群优化算法。这些算法通过引入混沌映射，对粒子的速度和位置更新策略以及更新公式进行了改进，从而提高了算法的全局搜索能力和局部开发能力。通过构建合适的混沌映射函数，将混沌运动的特性引入粒子群算法中，使得粒子能够更好的平衡搜索与开发的过程，避免过早收敛于局部最优解。同时，算法还通过自适应的参数，实现了对粒子运动轨迹以及搜索区域的灵活控制，进一步提高了算法的搜索效率。

为了验证算法的有效性，论文设计了一系列实验，并与传统的粒子群算法以及一些著名的粒子群算法的变体和著名的元启发式算法进行了对比。实验在CEC2005、CEC2014和CEC2017三个基准测试集上进行，所有算法在同等的实验条件下进行30次实验，并记录各个算法的均值、标准差和最优适应度。为保证实验结果的科学性，对所有算法的实验结果分别进行了Frideman秩检验，以及Wilcoxon秩和检验。实验结果表明，基于混沌映射的粒子群算法在解决复杂优化问题时表现出了更高的收敛精度。在对三个测试集的优化中，新算法分别在92.3%、66.7%和65.5%的测试中获得了冠军，并且在0.05的显著性水平上优于其他算法。此外，论文还优化了三个工程设计问题。通过在实际工程问题上与其他算法进行对照实验，展示了新算法在工程优化领域的优越性能和应用价值。这些应用案例不仅验证了算法的有效性，也为相关领域的优化问题提供了新的解决方案。

最后，论文总结了研究成果，指出了研究的不足和未来的发展方向。通过本文的研究，不仅丰富了粒子群算法的理论体系，也为混沌映射在优化算法中的应用提供了新的思路和方法。未来的研究可以进一步探索混沌映射与其他优化算法的融合，以及在更多领域的应用拓展。

The optimization and application of particle swarm optimization algorithm based on chaotic mapping have been systematically studied in this paper. The paper first provides a comprehensive and in-depth analysis of the basic principles of the PSO algorithm, and elaborates its optimization mechanism, advantages and disadvantages, as well as the limitations it faces when dealing with complex optimization problems. These limitations mainly include easy to fall into local optimal solutions, slow convergence speed, and limited ability to solve certain specific problems, which leads to lower solution accuracy. In order to overcome these shortcomings, the thesis further introduces the theory of chaotic mapping, discusses in depth the nonlinear, ergodic and stochastic properties of a variety of chaotic mappings, and analyzes the potential application value of these chaotic mapping methods in the design of optimization algorithms.

Based on the characteristics of various chaotic mappings, including Tent chaotic mapping and Logistic chaotic mapping, a new class of particle swarm optimization algorithms is proposed in this paper. These algorithms improve the velocity and position update strategies of particles as well as the update formulas by introducing chaotic mappings, which improves the algorithm's global search capability and local exploitation capability. By constructing a suitable chaotic mapping function and introducing the characteristics of chaotic motion into the particle swarm algorithms, the particles are able to better balance the process of searching and exploitation, and avoid converging to the local optimal solution prematurely. At the same time, the algorithm also realizes the flexible control of the particle motion trajectory and the search area through the adaptive parameters, which further improves the search efficiency of the algorithm.

In order to verify the effectiveness of the algorithm, the paper designs a series of experiments and compares it with the traditional particle swarm algorithm as well as some variants of the famous particle swarm algorithm and the famous meta-heuristic algorithm. The experiments are conducted on three benchmark test sets, CEC2005, CEC2014 and CEC2017, and all the algorithms are subjected to 30 experiments under the same experimental conditions, and the mean, standard deviation and optimal fitness of each algorithm are recorded. To ensure the scientific validity of the experimental results, Frideman rank test, and Wilcoxon rank sum test are performed on the experimental results of all algorithms respectively. The experimental results show that the particle swarm algorithm based on chaotic mapping exhibits higher convergence accuracy in solving complex optimization problems. In the optimization of three test sets, the new algorithm won 92.3%, 66.7%, and 65.5% of the tests, respectively, and outperformed the other algorithms at the 0.05 significance level. In addition, the paper optimizes three engineering design optimization problems. The superior performance and application value of the new algorithm in the field of engineering optimization are demonstrated through controlled experiments with other algorithms on real engineering problems. These application cases not only verify the effectiveness of the algorithm, but also provide new solutions for optimization problems in related fields.

Finally, the paper summarizes the research results and points out the shortcomings of the research and the future development direction. The research of this paper not only enriches the theoretical system of particle swarm algorithm, but also provides new ideas and methods for the application of chaotic mapping in optimization algorithm. Future research can further explore the integration of chaotic mapping with other optimization algorithms, as well as the expansion of applications in more fields.

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