Not All Chaos is Equal: An In-Depth Evaluation of Ten Chaotic Maps in Dholes-Inspired Optimization for Constrained Engineering Problems

Chaos-enhanced search mechanisms are widely used to improve exploration and mitigate stagnation in metaheuristic optimization. However, their integration into the recently proposed Dholes-Inspired Optimization (DIO) algorithm has not been previously studied. This work presents the first systematic evaluation of chaotic DIO variants, achieved by embedding ten structurally diverse chaotic maps into DIO and benchmarking their performance across seventeen classical constrained engineering design problems. Each chaotic map is first analysed through dynamical diagnostics—including time-series behaviour, cobweb structure, return maps, and sampling-density patterns—to assess its ergodicity and distributional characteristics. All algorithmic variants are then evaluated under uniform experimental conditions (identical FE budgets, 51 runs, and fixed seed scheduling). Results demonstrate that Piecewise, Singer, Iterative, and Circle maps consistently enhance DIO’s convergence reliability, producing the lowest mean ranks and highest win counts across the testbed. Statistical tests (Friedman, Nemenyi, Wilcoxon) confirm the significance of these improvements. Representative convergence curves show that chaotic variants accelerate early exploration and achieve feasible designs more reliably on complex, multi-constraint landscapes. Across the seventeen constrained engineering problems, four chaotic maps—Piecewise, Singer, Iterative, and Circle—consistently form a statistically superior elite group, offering the most reliable performance improvements. These results provide clear, evidence-based guidance for practitioners seeking effective chaos mechanisms within DIO and similar swarm optimizers. The code is publicly available on https://in.mathworks.com/matlabcentral/fileexchange/182678-not-all-chaos-is-equal.