Chaos is a random phenomenon in nonlinear dynamic systems, which is regular and random, and is sensitive to initial conditions and ergodicity. According to these characteristics, chaotic graphs represented by different equations are constructed to update the random variables in the optimization algorithm. Table 1 and Fig. 1 show ten chaotic maps which are used in the experiments. These ten chaotic maps have different effects in generating numerical values. More details about the 10 chaotic maps can be found in the literature [31,32]. Many researchers have demonstrated the effectiveness of chaotic maps in their studies, investigating the contribution of chaotic operators in the HHO [33], Krill Herd Algorithm (KHA) [34] and WOA [35].

Through experiments, it is found that the original SSA is easy to fall into the local optimum, which leads to premature convergence. As shown in Fig. 2b, each iteration update of its discoverer approaches the individual optimal solution straight line, which has a strong exploitation ability, but loses the exploration of the nearby search space in the process of approaching the optimal individual, the population diversity is reduced, and it is easy to fall into the local optimum. Therefore, we introduce a logarithmic spiral search model [21] to solve this problem. The mathematical model is described as follows:

(4)Xpbestt=|Xi,jt−Xpbestt|⋅eal⋅cos⁡(2πθ)+Xpbestt

(5)l=2(1−t/itermax)−1

where a is constant that determines the shape of the spiral, whose value is 1, l is a parameter that linearly decreases from 1 to −1, and Xpbest is the optimal position of the current iteration individual.

It can be seen from the Fig. 2a that when individuals of each generation update their positions, they gradually approach in a spiral shape, increasing the search for the surrounding space, maintaining the diversity of the population, and enhancing the exploration ability of the algorithm. Based on this analysis, the position update formula is adjusted as follows: (6)Xpbestt={Xpbestt×exp(−iα×itermax)if R2<ST & if R3<p|Xi,jt−Xpbestt|×eal×cos⁡(2πθ)+Xpbesttif R2<ST & if R3≥pXi,jt+Q×Lif R2≥ST where R3 is a uniformly distributed random number from 0 to 1, p is a constant and the value is 0.5.

In the SSA, two strategies are used for the location update of the guards. The Gaussian distribution is used to generate the step size for individuals with poor fitness. It can be seen from the Fig. 3 that the probability of the Gaussian distribution producing a smaller step size is higher. Conducive to the global search of the algorithm. The random step strategy is used for individuals with better fitness, and there is still a greater probability of large step in the later iterations, which is not conducive to algorithm convergence. Based on the above analysis, in order to balance the exploitation and exploration capabilities of the algorithm and enhance the convergence speed of the algorithm, an adaptive step size update formula is proposed for two strategies: (7)β={Cauchyrnd()if fitmeanSEL(t)≥fitmeanSEL(t+1)Gaussianrnd()if fitmeanSEL(t)<fitmeanSEL(t+1) (8)K=(2rand−1)×(1−t/itermax)1/2 (9)Xi,jt+1={Xbestt+β×|Xi,jt−Xbestt|if fi>fgXi,jt+K×(|Xi,jt−Bworstt|(fi−fw)+ε)if fi=fg where fitmeanSEL is the average fitness of the dominant population and SEL is the ratio of the dominant population which is 0.35.

For the individuals with poor fitness, when the dominant population of the updated sparrow is better than the dominant population of the previous generation, the larger step size of the Cauchy distribution is used to make the poor individual approach to the dominant population quickly; while when the dominant population of the updated sparrow is weaker than the dominant population of the previous generation, indicating that the renewal effect of this generation is not good, the smaller step size of Gaussian distribution is used to strengthen the search of the space near the individual. For individuals with better fitness, the adaptive step strategy is used. As can be seen from the Fig. 4, the large step size produced by the large probability in the early stage is beneficial for the individual to jump out of the local optimization, maintain the population diversity, increase the probability of small step size in the later stage, and impose only a small disturbance on the dominant individual, which is conducive to the convergence of the algorithm.

The pseudo code and flow chart of CLSSA is shown in Algorithm 2 and Fig. 5.

In this paper, 23 classical test functions are employed, including 7 unimodal functions, 6 multimodal functions and 10 fixed dimensional functions. The above test functions are all single-objective functions. The unimodal function F1–F7 has only one global optimal value, which is mainly used to test the development ability of the algorithm; the multimodal function has multiple local minima, which can be used to test the exploration ability of the algorithm. The benchmark function is shown in Table 2. The 3D view of each test function is shown in Figs. 6a–6d.

Ten kinds of chaotic maps are combined with SSA algorithm to form new algorithms, the first chaotic map combined algorithm is named SSA-1, the second chaotic map combined algorithm is named SSA-2, and so on. The ten combined algorithms are compared with SSA in the benchmark function. In order to make a fair comparison, on the same experimental platform, the number of populations is set to 50, and the maximum number of iterations is 300. Except for using chaotic sequences to replace parameter R2, the other parameters are consistent with the original literature, and the initial values of chaotic mapping sequences are set to 0.7. All the algorithms are implemented in MATLAB R2016a and the test environment is set up on a computer with AMD R7 4700U CPU@1.80 GHz 16GB RAM, running on Windows 10. The average value is used to measure the accuracy of the algorithm, and the standard deviation is used to measure the robustness of the algorithm, so the average value and standard deviation are used to measure the performance of the algorithm. The results are recorded in Table 3. The last row in this table presents the count of the better than, equal to or worse than SSA obtained by each chaotic map over all functions.

It can be observed from Table 3 that SSA-8 (Singer map) outperforms or equals SSA in 14 test functions. SSA-3 (Gauss map), SSA-9 (Sinusoidal map) and SSA-10 (Tent map) perform better than or equal to SSA in 15 test functions. SSA-5 (Logistic map), SSA-6 (Precewise map) and SSA-7 (Sine map) perform better than or equal to SSA in 16 test functions. SSA-1 (Chebyshev map), SSA-2 (Circle map) and SSA-4 (Iterative map) are superior or equal to SSA in 17 test functions.

In order to further analyze the optimization ability of the eleven algorithms, the results of these algorithms in each test function are compared and sorted according to the mean value of Table 3. The results are shown in Table 4, and the average sorting results of each algorithm in the last behavior of the table. SSA-4 ranks first, indicating that iterative mapping is the best alternative to the original parameter R2. In addition to SSA-4, SSA-2, SSA-3, SSA-5, and SSA-7 also rank higher than SSA. The above analysis shows that using chaotic map sequence to replace the random parameter R2 in the algorithm can better improve the algorithm's optimization performance, and each chaotic map sequence has different improvement effects on the algorithm. In order to visually show the performance of each algorithm in different test functions, a block diagram is used to plot the ranking results in Table 4, as shown in Fig. 7. The larger the area of the circle in the Fig. 7, the darker the color, indicating that the algorithm has a stronger performance in this test function, and the numbers in the figure indicate the ranking of each algorithm in each test function. It can be seen from Fig. 7 that SSA-4 performs better in the test function, and only performs poorly on F12 and F14.

In order to further prove the effectiveness of using chaotic sequences to replace SSA algorithm parameter, Figs. 8 and 9 list the convergence curves and box plots of eleven algorithms. It can be seen from Fig. 8 that SSA performs generally in each test function, and all chaotic mapping combination algorithms are better than SSA in convergence speed and convergence accuracy. The box diagram is used to show the distribution of the solutions of each algorithm. It can be seen from Fig. 9 that the optimal, median and worst values of the improved algorithm are better than those of SSA in most functions.

Combined with the above analysis, the chaotic mapping sequence can promote the improvement of SSA performance, and iterative mapping has the best effect on improving the performance of the SSA. Therefore, in the next part of the CLSSA performance test, the iterative mapping sequence is used to replace the random value parameter R2 in the SSA.

As mentioned above, this paper mainly uses three strategies to improve SSA, so three different derivative algorithms are designed to evaluate the impact of these three strategies on the algorithm. These three derivation algorithms are obtained by removing the corresponding improvement strategy from CLSSA. CLSSA-1 removes both logarithmic spiral strategy and adaptive step strategy; CLSSA-2 removes chaotic map and adaptive step strategy at the same time; CLSSA-3 removes chaotic map strategy and logarithmic spiral strategy at the same time. 23 benchmark functions are used to compare the performance of the three derived algorithms with SSA and CLSSA. Each algorithm runs 30 times independently on each test function, and the statistical average results are shown in Tables 5 and 6 show the ranking of each algorithm in the test function. Obviously, CLSSA which includes all the improvement strategies, performed best, ranking first on average. The performance of the derived algorithm with one improved strategy is better than that of SSA. From the specific optimization results and sorting table provided by the Table 6, the chaotic mapping strategy mainly improves the development ability, while the logarithmic spiral strategy enhances the exploration ability of the algorithm, while the adaptive step strategy enhances the exploitation ability and exploration ability of the algorithm to some extent. The above analysis proves the effectiveness of each improvement strategy.

In order to verify the performance of CLSSA, the proposed CLSSA is compared with the SSA, WOA, BSO [36], PSO, GSA, HHO, GWO [37], SCA [38], MVO [39], MFO [40], BBO, FPA [41] Flower pollination algorithm for global optimization. The experimental environment is the same as the previous article, the number of populations is set to 50, the maximum number of iterations is 300, and the parameters of each algorithm are consistent with the original literature. Meanwhile, to reduce the influence of randomness on the experimental results, all algorithms need to run 30 times independently. Table 7 lists the best fitness, mean fitness and standard deviation, in which the best mean fitness is marked in bold.

As shown in Table 7, when solving the unimodal test functions F1–F7, CLSSA can stably converge to the optimal value in F1–F4 and F6, and the performance is better than the comparison algorithm. CLSSA could not obtain the optimal value of F5, but it was 19 orders of magnitude higher than SSA. HHO perform best on F7, with CLSSA in second position. In the unimodal test functions F1–F7, CLSSA is better than SSA, indicating that the proposed chaotic map sequence substitution strategy can effectively improve the local search ability of the algorithm.

When solving the multimodal test functions F8–F13, GSA, PSO, BBO and MFO outperform CLSSA in solving F8. For F9–F11, CLSAA, SSA, HHO can all stably converge to the optimal value. WOA can obtain the optimal value, but it is not stable. The CLSSA has the highest accuracy for F12–F13, with optimal values improved by 17 and 11 orders of magnitude compared to SSA. When solving the fixed-dimensional multimodal functions F14–F23, the CLSSA performs poorly for F14, outperforming only SSA, WOA, GSA, GWO and BBO. For the F15, optimal values can be obtained for CLSSA and SSA, but CLSSA is more stable than SSA. All algorithms have similar performance at F16, and all can obtain optimal values. GSA is the most stable and CLSAA is the second most stable. The CLSSA outperforms WOA, HHO, GSA, CSA, MVO, BBO and FPA for F17, with performance comparable to other algorithms. As for F18, the stability of CLSSA is only weaker than BSO, PSO, and GSA. The CLSSA outperforms all comparison algorithms for F19, F21 and F23. The GSA performs best for F22, with the CLSSA second best. In all multimodal test functions, CLSSA performs better than SSA, which shows that the logarithmic spiral strategy proposed in this paper can significantly improve the performance of algorithm exploration.

Combined with the above analysis, the CLSSA proposed in this paper is better than all the comparison algorithms in 12 of the 23 benchmark functions, 11 comparison algorithms in 6 test functions, 9 comparison algorithms in 3 test functions, and CLSSA is better than SSA, in all test functions, which proves that our proposed CLSSA has obvious advantages in optimization accuracy.

In order to directly show the performance differences of each algorithm in solving the test function, the algorithms are sorted according to the mean fitness of Table 7, the results are shown in Table 8, and the last column is the average ranking of each algorithm.

Fig. 10 is drawn according to the ranks in Table 8. The smaller the area of the algorithm performance curve, the better the performance of the algorithm.

The black bold line is the sorting result curve of CLSSA, and it can be seen intuitively that the performance of CLSSA is in the middle level on F8 and F14, and performs better in other test functions, and its surrounding area is the smallest, indicating that CLSSA has the best optimization performance as a whole.

To further illustrate the convergence performance of CLSSA, Fig. 11 lists the mean convergence curves of 13 algorithms to solve these functions. For the unimodal test function F1–F7, CLSSA has the best performance, the convergence speed is faster than all comparison algorithms, and the convergence accuracy is also higher than all comparison algorithms. For the multimodal functions F8–F23, CLSSA performs best in most of them. However, for functions F14, F17, and F18, CLSSA converges slowly in the early iterations. The CLSSA converges slower in the early iterations on F21 and F22 but can converge to better results afterwards. CLSSA converges faster than SSA on all test functions, which shows that the variable step strategy proposed in this paper can effectively improve the convergence speed.

To analyze the distribution characteristics of each algorithm in the test function, Fig. 12 lists box plots of 13 algorithms. Compared with other comparison algorithms, the CLSSA proposed in this paper performs well on most functions, and the obtained maximum, minimum, and median values are almost the same as the optimal solution, especially for F9, F10, F11, F16, F17, F19 and F20. In other test functions, although there are individual outliers, the overall distribution is still more concentrated than the comparison algorithm. Therefore, the CLSSA proposed in this paper has stronger stability.

The above analysis shows that CLSSA shows strong optimization ability on low-dimensional functions. However, the optimization algorithm is prone to fail in solving high-dimensional complex function problems. Real-world optimization problems are mostly large-scale complex optimization problems. Therefore, to verify the performance of CLSSA in high-dimensional problems, 13 algorithms were compared on the 100D test functions, and the experimental results are shown in Table 9. CLSSA is better than all comparison algorithms in F1–F6 and F12–F13. HHO performs best in F7 and F8, CLSSA ranks second and third respectively. When solving F9–F11, CLSSA and SSA can get the best value. It can also be seen from Figs. 13 and 14 that CLSSA performs well in other test functions except for the pool performance on F8 and can steadily and quickly converge to a better value.

In summary, compared with other algorithms, the CLSSA proposed in this paper is competitive, and the proposed improvement strategy can handle the relationship between exploitation and exploration well.

Engineering design problem is a nonlinear optimization problem with complex geometric shapes, various design variables and many practical engineering constraints. The performance of the proposed algorithm is evaluated by solving practical engineering problems. In the simulation, the population size is set to 50, and the maximum iterations is 500. The results of 30 independent runs of CLSSA are compared with those in other literatures.