MATLAB求一个Rossler微分方程组……

一般用for循环+num2str就可以实现

我写的程序 结论是无解

function b=sysolve()

warning off

bb=size(0:0.01:0.65)

b=cell(bb)

for ii=0:0.01:0.65

a=['Dy=x+',num2str(ii),'*y']

kk=dsolve('Dx=-y',a,'Dz=2+z*(x-4)')

if ~isempty(kk)

b{ii}.ans=kk

b{ii}.i=ii

end

end

end

% Author: Thomas Lee

% E-mail: lixf1979@126.com % Corresponding: School of Mathematics, Physics

% and Software Engineering, Lanzhou Jiaotong University, Lanzhou 730070, China

function dX = Rossler_ly(t,X)

% Rossler吸引子，用来计算Lyapunov指数

%a=0.15,b=0.20,c=10.0

%dx/dt = -y-z,

%dy/dt = x+ay,

%dz/dt = b+z(x-c),

a = 0.15

b = 0.20

c = 10.0

x=X(1)y=X(2)z=X(3)

% Y的三个列向量为相互正交的单位向量

Y = [X(4), X(7), X(10)

X(5), X(8), X(11)

X(6), X(9), X(12)]

% 输出向量的初始化，必不可少

dX = zeros(12,1)

% Rossler吸引子

dX(1) = -y-z

dX(2) = x+a*y

dX(3) = b+z*(x-c)

% Rossler吸引子的Jacobi矩阵

Jaco = [0 -1 -1

1 a 0

z 0 x-c]

dX(4:12) = Jaco*Y

求解LE代码：

% 计算Rossler吸引子的Lyapunov指数

clear

yinit = [1,1,1]

orthmatrix = [1 0 0

0 1 0

0 0 1]

a = 0.15

b = 0.20

c = 10.0

y = zeros(12,1)

% 初始化输入

y(1:3) = yinit

y(4:12) = orthmatrix

tstart = 0% 时间初始值

tstep = 1e-3% 时间步长

wholetimes = 1e5% 总的循环次数

steps = 10% 每次演化的步数

iteratetimes = wholetimes/steps% 演化的次数

mod = zeros(3,1)

lp = zeros(3,1)

% 初始化三个Lyapunov指数

Lyapunov1 = zeros(iteratetimes,1)

Lyapunov2 = zeros(iteratetimes,1)

Lyapunov3 = zeros(iteratetimes,1)

for i=1:iteratetimes

tspan = tstart:tstep:(tstart + tstep*steps)

[T,Y] = ode45('Rossler_ly', tspan, y)

% 取积分得到的最后一个时刻的值

y = Y(size(Y,1),:)

% 重新定义起始时刻

tstart = tstart + tstep*steps

y0 = [y(4) y(7) y(10)

y(5) y(8) y(11)

y(6) y(9) y(12)]

%正交化

y0 = ThreeGS(y0)

% 取三个向量的模

mod(1) = sqrt(y0(:,1)'*y0(:,1))

mod(2) = sqrt(y0(:,2)'*y0(:,2))

mod(3) = sqrt(y0(:,3)'*y0(:,3))

y0(:,1) = y0(:,1)/mod(1)

y0(:,2) = y0(:,2)/mod(2)

y0(:,3) = y0(:,3)/mod(3)

lp = lp+log(abs(mod))

%三个Lyapunov指数

Lyapunov1(i) = lp(1)/(tstart)

Lyapunov2(i) = lp(2)/(tstart)

Lyapunov3(i) = lp(3)/(tstart)

y(4:12) = y0'

end

% 作Lyapunov指数谱图

i = 1:iteratetimes

plot(i,Lyapunov1,i,Lyapunov2,i,Lyapunov3)

程序中用到的ThreeGS程序如下：

%G-S正交化

function A = ThreeGS(V) % V 为3*3向量

v1 = V(:,1)

v2 = V(:,2)

v3 = V(:,3)

a1 = zeros(3,1)

a2 = zeros(3,1)

a3 = zeros(3,1)

a1 = v1

a2 = v2-((a1'*v2)/(a1'*a1))*a1

a3 = v3-((a1'*v3)/(a1'*a1))*a1-((a2'*v3)/(a2'*a2))*a2

A = [a1,a2,a3]

计算得到的Rossler系统的LE为———— 0.063231 0.092635 -9.8924