混沌映射使用_sine混沌映射

 混沌映射被用于生成混沌序列，这是一种由简单的确定性系统产生的随机性序列。一般混沌序列具有以下主要特征：非线性；对初值的敏感依赖性；遍历性；随机性；奇异吸引子（混沌吸引子）；分数维持性；整体稳定局部不稳定；长期不可预测性；轨道不稳定性及分叉；普适性和Feigenbaum常数。由于目前大多数智群能算法在初始化阶段都是随机生成，因此很多研究者将混沌映射应用于种群初始化，以增加算法的随机性和多样性。本文介绍了以下五种混沌映射，包括：Logistic映射、Circle映射、Sine映射、Singer映射和Cubic映射，并将以上映射方式应用于鲸鱼优化算法中，若需要应用在其他算法，只需替换初始化函数即可。

一、运行结果
Logistic映射结果

The best solution obtained by WOA is : 3.999      4.0162      4.0002       3.985The best optimal value of the objective funciton found by WOA is : -10.3575
二、部分Matlab代码
clc
clear
close all
 
SearchAgents_no=30; % Number of search agents
%种群数量
 
Function_name='F22'; % Name of the test function that can be from F1 to F23 (Table 1,2,3 in the paper)
%使用方程的名字，对应Functions_details 文件
 
Max_iteration=500; % Maximum numbef of iterations 
%最大迭代次数
 
% Load details of the selected benchmark function
[lb,ub,dim,fobj]=Get_Functions_details(Function_name);%得到具体的方程即目标函数，变量的维度
 
[Best_score,Best_pos,WOA_cg_curve]=WOA(SearchAgents_no,Max_iteration,lb,ub,dim,fobj);
%最小值，优化每个维度x的值，迭代曲线 
 
figure('Position',[269   240   660   290])
%Draw search space
subplot(1,2,1);
func_plot(Function_name);
title('Parameter space')
xlabel('x_1');
ylabel('x_2');
zlabel([Function_name,'( x_1 , x_2 )'])
 
%Draw objective space
subplot(1,2,2);
semilogy(WOA_cg_curve,'Color','r')
title('Objective space')
xlabel('Iteration');
ylabel('Best score obtained so far');
 
axis tight
grid on
box on
legend('WOA')
 
display(['The best solution obtained by WOA is : ', num2str(Best_pos)]);
display(['The best optimal value of the objective funciton found by WOA is : ', num2str(Best_score)]);
 
function [Leader_score,Leader_pos,Convergence_curve]=WOA(SearchAgents_no,Max_iter,lb,ub,dim,fobj)
 
% initialize position vector and score for the leader
Leader_pos=zeros(1,dim);
Leader_score=inf; %change this to -inf for maximization problems
 
 
%Initialize the positions of search agents
Positions=initialization(SearchAgents_no,dim,ub,lb);
 
Convergence_curve=zeros(1,Max_iter);
 
t=0;% Loop counter
 
% Main loop
while t<Max_iter
    for i=1:size(Positions,1)
        
        % Return back the search agents that go beyond the boundaries of the search space
        Flag4ub=Positions(i,:)>ub;
        Flag4lb=Positions(i,:)<lb;
        Positions(i,:)=(Positions(i,:).*(~(Flag4ub+Flag4lb)))+ub.*Flag4ub+lb.*Flag4lb;
        
        % Calculate objective function for each search agent
        fitness=fobj(Positions(i,:));
        
        % Update the leader
        if fitness<Leader_score % Change this to > for maximization problem
            Leader_score=fitness; % Update alpha
            Leader_pos=Positions(i,:);
        end
        
    end
    
    a=2-t*((2)/Max_iter); % a decreases linearly fron 2 to 0 in Eq. (2.3)
    
    % a2 linearly dicreases from -1 to -2 to calculate t in Eq. (3.12)
    a2=-1+t*((-1)/Max_iter);
    
    % Update the Position of search agents 
    for i=1:size(Positions,1)
        r1=rand(); % r1 is a random number in [0,1]
        r2=rand(); % r2 is a random number in [0,1]
        
        A=2*a*r1-a;  % Eq. (2.3) in the paper
        C=2*r2;      % Eq. (2.4) in the paper
        
        
        b=1;               %  parameters in Eq. (2.5)
        l=(a2-1)*rand+1;   %  parameters in Eq. (2.5)
        
        p = rand();        % p in Eq. (2.6)
        
        for j=1:size(Positions,2)
            
            if p<0.5   
                if abs(A)>=1
                    rand_leader_index = floor(SearchAgents_no*rand()+1);
                    X_rand = Positions(rand_leader_index, :);
                    D_X_rand=abs(C*X_rand(j)-Positions(i,j)); % Eq. (2.7)
                    Positions(i,j)=X_rand(j)-A*D_X_rand;      % Eq. (2.8)
                    
                elseif abs(A)<1
                    D_Leader=abs(C*Leader_pos(j)-Positions(i,j)); % Eq. (2.1)
                    Positions(i,j)=Leader_pos(j)-A*D_Leader;      % Eq. (2.2)
                end
                
            elseif p>=0.5
              
                distance2Leader=abs(Leader_pos(j)-Positions(i,j));
                % Eq. (2.5)
                Positions(i,j)=distance2Leader*exp(b.*l).*cos(l.*2*pi)+Leader_pos(j);
                
            end
            
        end
    end
    t=t+1;
    Convergence_curve(t)=Leader_score;
end
end
function Positions=initialization(SearchAgents_no,dim,ub,lb)
Boundary_no= size(ub,2); % numnber of boundaries
Positions=zeros(SearchAgents_no,dim);HUN