Natural systems often exhibit chaotic behavior in their space-time evolution. Systems transiting between chaos and order manifest a potential to compute, as shown with cellular automata and artificial neural networks. We demonstrate that swarm optimization algorithms also exhibit transitions from chaos, analogous to a motion of gas molecules, when particles explore solution space disorderly, to order, when particles follow a leader, similar to molecules propagating along diffusion gradients in liquid solutions of reagents. We analyze these ‘phase-like’ transitions in swarm optimization algorithms using recurrence quantification analysis and Lempel-Ziv complexity estimation. We demonstrate that converging iterations of the optimization algorithms are statistically different from non-converging ones in a view of applied chaos, complexity and predictability estimating indicators. An identification of a key factor responsible for the intensity of their phase transition is the main contribution of this paper. We examined an optimization as a process with three variable factors—an algorithm, number generator and optimization function. More than 9000 executions of the optimization algorithm revealed that the nature of an applied algorithm itself is the main source of the phase transitions. Some of the algorithms exhibit larger transition-shifting behavior while others perform rather transition-steady computing. These findings might be important for future extensions of these algorithms.

中文翻译：

---

群优化算法中的扰动和相变

自然系统在其时空演化中通常表现出混沌行为。如细胞自动机和人工神经网络所示，在混沌和有序之间转换的系统具有计算的潜力。我们证明，当粒子无序地探索溶液空间时，群体优化算法还表现出从混沌到气体分子运动的过渡，到粒子跟随前导时有序的过渡，类似于沿着试剂液体溶液中的扩散梯度传播的分子。我们使用递归量化分析和Lempel-Ziv复杂度估算，在群体优化算法中分析了这些“相态”过渡。我们证明，从应用混沌的角度来看，优化算法的收敛迭代在统计上与非收敛迭代不同，复杂性和可预测性估计指标。识别导致其相变强度的关键因素是本文的主要贡献。我们将优化视为具有三个可变因素的过程-算法，数字生成器和优化函数。优化算法的9000多次执行表明，所应用算法本身的性质是相变的主要来源。一些算法表现出较大的过渡转移行为，而其他算法则执行相当稳定的过渡计算。这些发现对于将来扩展这些算法可能很重要。我们将优化视为具有三个可变因素的过程-算法，数字生成器和优化函数。优化算法的9000多次执行表明，所应用算法本身的性质是相变的主要来源。一些算法表现出较大的过渡转移行为，而其他算法则执行相当稳定的过渡计算。这些发现对于将来扩展这些算法可能很重要。我们将优化视为具有三个可变因素的过程-算法，数字生成器和优化函数。优化算法的9000多次执行表明，所应用算法本身的性质是相变的主要来源。一些算法表现出较大的过渡转移行为，而其他算法则执行相当稳定的过渡计算。这些发现对于将来扩展这些算法可能很重要。