DMD algo

clear all, close all, clc

xi=linspace(-10,10,400);
t=linspace(0,100*pi,200);
dt=t(2)-t(1);
[Xgrid,T]=meshgrid(xi,t);

%create two spatio-temporal patterns
f1=sech(Xgrid+3).*(1*exp(i*2.3*T));
f2=(sech(Xgrid).*tanh(Xgrid)).*(2*exp(i*2.8*T));
f=f1+f2;
[u,s,v]=svd(f.');
subplot(2,2,1), surfl(Xgrid,T,real(f1)); shading interp, colormap(gray)
subplot(2,2,2), surfl(Xgrid,T,real(f2)); shading interp, colormap(gray)
subplot(2,2,3), surfl(Xgrid,T,real(f2)); shading interp, colormap(gray)

figure(2)
plot(diag(s)/(sum(diag(s))),'ro');

figure(3)
subplot(2,1,1), plot(real(u(:,1:2)))
subplot(2,1,2), plot(real(v(:,1:2)))

%% DMD ALGORITHM
X=f.';
X1=X(:,1:end-1);
X2=X(:,2:end);
r=2;
[U,S,V]=svd(X1,'econ');
Ur=U(:,1:r);
Sr=S(1:r,1:r);
Vr=V(:,1:r);

Atilde=Ur'*X2*Vr/Sr;

[W,D]=eig(Atilde);
Phi=X2*Vr/Sr*W; %DMD modes

lambda=diag(D);
omega=log(lambda)/dt;
figure(3),
subplot(2,1,1),hold on,plot(real(Phi))

x1=X(:,1);
b=Phi\x1;
time_dynamics=zeros(r,length(t));
for iter=1:length(t)
    time_dynamics(:,iter)=(b.*exp(omega*t(iter)));
end
X_dmd=Phi*time_dynamics;
figure(1)
subplot(2,2,4),surfl(Xgrid,T,real(X_dmd).'); shading interp, colormap(gray)