## 1 Introduction
Parameter adaptation eliminates the need for experts to be aware of the specifics of optimization problems by automatically updating the control parameters to suitable values. Results from simulations demonstrate that JADE is superior to both traditional DE algorithms and adaptive variants. Built upon the JADE, the researchers developed a parameter adaptive DE algorithm (SHADE) that stores the parameters used in previous successful evolution rounds [34]. jDE (self-adaptive tuning parameters) is another well-known variant of DE algorithm that has shown acceptable performance on benchmark problems, which uses an evolution process to fine tune the optimization algorithm parameters [35]. BOA is a novel optimization technique that is based on the foraging and mating behaviour of butterflies [21]. As they search for food, butterflies emit a fragrance, the intensity of which is proportional to the abundance of food sources. Other butterflies in the swarm can detect the strength of the fragrance released by other butterflies to estimate the likely location of a food supply or mate. Therefore, the BOA algorithm imitates feeding and mating habits of biological butterflies, assuming that a butterfly would produce a fragrance of varying strength, which is associated with the butterfly's objective function value. The fragrance will propagate over a distance, and other butterflies can sense it. This is how the butterflies can share their personal information with other butterflies and form a collective social knowledge network. Due to its simplicity and effectiveness, BOA has been suggested to solve a variety of engineering applications, from WSN sensor localization [29], engineering design problems [36], and neural network training [30], etc. Despite the fact that the BOA has shown better performance in real-world applications compared to other meta-heuristic algorithms, it also faces many challenging issues. For instance, the basic BOA is easy to fall into the local optimum when solving highly complex problems, and the convergence speed will reduce greatly in the late iteration process [37]. Therefore, many improved variants of BOA have been suggested to overcome the two aspects [36; 38]. In order to further improve the optimization performance on complex optimization problems, researchers resorted to hybridizing two or more algorithms, thus creating one new algorithm that exploits the complementary strengths of each method. Some examples of the combination of two meta-heuristic algorithms include: the HFPSO [39] algorithm, where the FA and PSO algorithms are hybridized with the aim of increasing accuracy and improving convergence; a DE/WOA [26] hybrid algorithm, which combines local search abilities of WOA with DE global search; HPSOBOA [40], which combines BOA and PSO algorithms; etc. In a recent study, it was proposed to combine the butterfly optimization algorithm (BOA) and differential evolution (DE), named HBODEA [41], in order to more effectively utilise each method's advantages. Two enhancements--mutation and crossover operations of DE and dynamic adaptive operators--are added to the original BOA in the proposed identification strategy to increase global search capability and decrease the risk of being trapped in a local optimum. However, this hybridization left a lot of different improvements and variations to be performed. As a result, all of these papers demonstrate that the hybridization of EAs speeds up the process of achieving the best solution, prevents the issue of premature convergence, and yields better answers. Numerous applications of nonlinear dynamics, especially those incorporating chaos, have gained prominence as a consequence of the development of nonlinear dynamics and are now commonly employed to enhance the effectiveness of various optimization techniques [42]. The infinitely unstable periodic motions that characterise the bounded dynamic behaviour of non-linear systems is characterised as chaos [43].