非线性分析作业3.docxVIP

非线性分析第三次作业学 院（系）： 电子信息与电气工程学部专 业： 信号与信息处理 学 生 姓 名：代 菊 学 号 任 课 教 师：梅建琴大连理工大学Dalian University of TechnologyGiven the ODE: Plot the bifurcation diagram and phase diagrams as F varies, and investigate the routes to chaos.Compute the Lyapunov exponents, and plot the value as a function of F.答：1）令，上述微分方程可以化为：Matlab 程序代码如下：%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 定义ODE方程%%%%%%%%%%%%%%%%%%%%%%%%%%%function dx=ode(ignore,X)global F wd;r=1;x=X(1);v=X(2);psi=X(3);dx=zeros(3,1);dx(1)=v;dx(2)=-r*v+x-x^3+F*cos(psi);dx(3)=wd;%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%分岔图绘制程序%%%%%%%%%%%%%%%%%%%%%%%%%%%%function duffing_bifur_Fclear;clc;global F wd;wd=1.2;range=0.4:0.0001:0.47;%F的范围% range=0.4:0.001:0.47;%F的范围period=2*pi/wd;k=0;YY1=[];rangelength=length(range);YY1=ones(rangelength,3000)*NaN;step=2*pi/300/wd; %步长,由于wd=1，周期即为2*pi，此步长为1周期取100个点。for F=range y0=[2 0 0]; k=k+1; %除去前面60个周期的数据，并将最后的结果作为下一次积分的初值 tspan=0:step:60*period; [ignore,Y]=ode45(@duffing,tspan,y0); y0=Y(end,:); j=1; kkk=300; for ii=20:59 for point=(ii-1)*kkk+2:ii*kkk ifY(point,1)Y(point-2,1)Y(point,1)Y(point+2,1)Y(point,1)Y(point-100,1) YY1(k,j)=Y(point,1); j=j+1; end end %取出每一个周期内的第一个解的最后一个值。 y0=Y(end,:); endendplot(range,bifdata,k.,markersize,5);运行上述程序，并对结果进行分析：以F为自变量，运动幅度为因变量的分岔图如下：其混沌道路描述如下：当时，微分方系统为单周期运动，此时的相图如下所示：(b)当时，单摆处于双周期运动状态，此时的相图如下所示：(c)当，单摆经历倍周期分岔，此时相图如下所示(d) 当时，单摆进入混沌运动区，此时的系统相图如下所示：由该相图可知，系统在数个周期内作运动。(e) 当时，系统恢复规则运动，此时相图如下：由上图可知，系统从混沌中恢复，且做单周期运动。(2)wolf算法来计算李雅普诺夫指数的matlab程序如下：%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 杜芬方程的参数%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%function f=duff_ext(t,X);global F;r=1;x=X(1);y=X(2);psi=X(3);dx=zeros(3,1);f(1)=y;f(2)=-r*y+x-x^3+F*cos(psi);f(3)=0.2;%Linearized system.Jac=[0 , 1, 0; 1-3*x^2, -r, -F*sin(psi); 0, 0, 0];f(4:12)=Jac*Y;%变量方程%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 计算李雅普诺夫指数谱函数%%%%%%%%%%%%%%%%%%%%%%%%%%%%%functio