## 2 Materials and Methods
### Chaotic Ant Colony Optimization Algorithm  
A chaotic ant colony optimization (CACO) algorithm is a combination of chaos search and the ant colony optimization algorithm. After the ant colony search is completed, chaos is used to improve the search accuracy and avoid falling into the local optimum [28; 29; 30]. Therefore, the CACO algorithm takes on a global optimization ability. In addition, some new optimization algorithms have been proposed in recent years [41; 42; 43; 44; 45; 46; 47; 48; 49; 50; 51; 52; 53; 54; 55].  
Pheromones are substances secreted by an individual outside the body and detected by other individuals of the same species through olfactory organs, so that the latter shows some changes in behavior, emotion, psychological, or physiological mechanisms [56; 57; 58]. Since the selection of stochastic resonance system parameters plays an important role and the ACO has the same pheromone on each path during initialization, the pheromone update formula can be expressed as follows:  
\[\tau_{ij}(t+1)=(1-\rho)\tau_{ij}(t)+\Delta\tau_{ij}(t) \tag{7}\]  
where \(\tau_{ij}(t)\) represents the pheromone quantity on branch \(j\) at time \(t\). When \(t=0\), the information intensity on each path is equal, and \(\rho\) represents the volatile factor of the global pheromone, generally valued as [0, 1], which determines the volatilization speed of the pheromone. When each ant is looking for a path, has the same probability, and the searching period is long, the CACO is used to optimize the parameters of the stochastic resonance system, and the ergodicity of the chaos algorithm is introduced to initialize chaos. The iteration formula is described as follows:  
\[Z_{ij}(t+1)=\mu Z_{ij}(t)\big{[}1-Z_{ij}(t)\big{]} \tag{8}\]  
where \(\mu\) is the control parameter, and its value is [3.56, 4]. When \(\mu=4,\ 0\leq Z_{ij}\leq 1\), logistics mapping is completely chaotic. In order to avoid local optimization problems in the process of ACO, chaos disturbance was added to the process of pheromone adjustment so as to realize the whole optimization [59, 60, 61]. The pheromone adjustment of the CACO algorithm is described as follows:  
\[\tau_{ij}(t+1)=(1-\rho)\tau_{ij}(t)+\Delta\tau_{ij}(t)+q_{1}Z_{ij}(t) \tag{9}\]  
where \(Z_{ij}(t)\) is a chaotic variable, which is obtained by iteration of Equation (8), and \(q_{1}\) is a coefficient.  
The specific steps of the chaotic ACO are described as follows.  
* **Step 1.** Set the initial parameter and the maximum iteration N\(\mathrm{\acute{e}}\)max and generate a group of chaotic variables corresponding to the parameters to make the pheromone chaos.
* **Step 2.** Establish the solution space, put in a random ant, and establish the tabu table (Tabuk). The ant is now walking on a collection of points, and the initial point creates the allowed table, that is, the accessible points.
* **Step 3.** Each ant randomly selects the next point to visit; record each point in Tabuk. Update the pheromone and continue the process until all points within the set parameter range are recorded in Tabuk.
* **Step 4.** After all the ants have walked each point for the first time, record the current best result.
* In order to avoid local optimization of the parameters, use Equation (9) to update the pheromones.
* **Step 5.** When a number of iterations is completed, the algorithm stops. Otherwise, clear Tabuk and go to step 3 to recalculate. When there is no better result after several iterations, stop the algorithm and output the optimal value.