Lyapunov exponent for fractional order differential equation

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ABHIJIT MAJUMDER
ABHIJIT MAJUMDER 2020 年 5 月 4 日
編集済み: Walter Roberson 2022 年 7 月 26 日
Hi, I have three dimensional fde model of which I want to compute Lyapunov exponent with respect to parameter. I have taken this code from the paper entitled 'Matlab code for Lyapunov exponents of fractional order systems' but it's not working, Please help.
function [t,LE]=FO_Lyapunov_q(ne,ext_fcn,t_start,h_norm,t_end,x_start,h,q,p);
x=zeros(ne*(ne+1),1);
x0=x;
c=zeros(ne,1);
gsc=c; zn=c;
n_it = round((t_end-t_start)/h_norm);
x(1:ne)=x_start;
i=1;
while i<=ne
x((ne+1)*i)=1.0;
i=i+1;
end
t=t_start;
it=1;
while it<=n_it
[T,Y] = fde12(q,ext_fcn,t,t+h_norm,x,h);
t=t+h_norm;
Y=transpose(Y);
x=Y(size(Y,1),:);
i=1;
while i<=ne
j=1;
while j<=ne;
x0(ne*i+j)=x(ne*j+i);
j=j+1;
end;
i=i+1;
end;
zn(1)=0.0;
j=1;
while j<=ne
zn(1)=zn(1)+x0(ne*j+1)*x0(ne*j+1);
j=j+1;
end;
zn(1)=sqrt(zn(1));
j=1;
while j<=ne
x0(ne*j+1)=x0(ne*j+1)/zn(1);
j=j+1;
end
j=2;
while j<=ne
k=1;
while k<=j-1
gsc(k)=0.0;
l=1;
while l<=ne;
gsc(k)=gsc(k)+x0(ne*l+j)*x0(ne*l+k);
l=l+1;
end
k=k+1;
end
k=1;
while k<=ne
l=1;
while l<=j-1
x0(ne*k+j)=x0(ne*k+j)-gsc(l)*x0
(ne*k+l);
l=l+1;
end
k=k+1;
end;
zn(j)=0.0;
k=1;
while k<=ne
zn(j)=zn(j)+x0(ne*k+j)*x0(ne*k+j);
k=k+1;
end
zn(j)=sqrt(zn(j));
k=1;
while k<=ne
x0(ne*k+j)=x0(ne*k+j)/zn(j);
k=k+1;
end
j=j+1;
end
% update running vector magnitudes
k=1;
while k<=ne;
c(k)=c(k)+log(zn(k));
k=k+1;
end;
% normalize exponent
k=1;
while k<=ne
LE(k)=c(k)/(t-t_start);
k=k+1;
end
i=1;
while i<=ne
j=1;
while j<=ne;
x(ne*j+i)=x0(ne*i+j);
j=j+1;
end
i=i+1;
end;
x=transpose(x);
it=it+1;
end
%----------------------------------------------------------------------------------------------------------------------------%
function f=LE_RF(t,x,p)
f=zeros(size(x));
X= [x(4), x(7), x(10);
x(5), x(8), x(11);
x(6), x(9), x(12)];
%RF equations
f(1)=x(2)*(x(3)-1+x(1)*x(1))+0.1*x(1);
f(2)=x(1)*(3*x(3)+1-x(1)*x(1))+0.1*x(2);
f(3)=-2*x(3)*(p+x(1)*x(2));
%Jacobian matrix
J=[2*x(1)*x(2)+0.1, x(1)*x(1)+x(3)-1, x(2);
-3*x(1)*x(1)+3*x(3)+1,0.1,3*x(1);
-2*x(2)*x(3),-2*x(1)*x(3),-2*(x(1)*x(2)+p)];
%Righthand side of variational equations
f(4:12)=J*X; % To be modified if ne>3
%----------------------------------------------------------------------------------------------------------------------------%
function run_Lyapunov_p(ne,ext_fcn,t_start,h_norm,t_end,x_start,h,q,p_min,p_max,n);
hold on;
p_step=(p_max-p_min)/n
p=p_min;
while p<=p_max
[t,LE]=FO_Lyapunov_q(ne,ext_fcn,t_start,h_norm,t_end,x_start,h,q,p);
p=p+p_step
plot(p,LE);
end
end
%----------------------------------------------------------------------------------------------------------------------------%
run_FO_Lyapunov_q(3,@LE_RF,0,0.05,150,[0.1;0.1;0.1],0.002,0.9,1,800)

回答 (1 件)

nune pratyusha
nune pratyusha 2022 年 7 月 26 日
you have to download fde12.m file and put all programs in one folder then run. It will work

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