-
Notifications
You must be signed in to change notification settings - Fork 1
/
Duffing_example.m
140 lines (112 loc) · 4.39 KB
/
Duffing_example.m
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
clear
close all
addpath(genpath('./saved_data_from_runs'))
% NB general code for ResDMD can be found at: https://github.com/MColbrook/Residual-Dynamic-Mode-Decomposition
%% UNCOMMENT TO RUN
% rng(1)
% %% Set parameters
% M1=10^3; % number of data points
% M2=50;
% delta_t=0.25; % time step
% ODEFUN=@(t,y) [y(2);y(1)-y(1).^3];
% options = odeset('RelTol',1e-12,'AbsTol',1e-12);
% N=100;
% PHI = @(r) exp(-r); % radial basis function
%
% x_pts = -1.2:0.04:1.2; y_pts = -0.05:0.04:1.2;
% v=(10.^(-2:0.1:1));
%
% %% Produce the data
% X=[];
% Y=[];
% for jj=1:M1
% Y0=(rand(2,1)-0.5)*4;
% [~,Y1]=ode45(ODEFUN,[0 0.000001 (1:(3+M2))*delta_t],Y0,options);
% Y1=Y1';
% X = [X,Y1(:,[1,3:M2+1])];
% Y = [Y,Y1(:,3:M2+2)];
% end
% M = M1*M2;
%
% d=mean(vecnorm(X-mean(X')')); % scaling for radial function
%
% [~,C] = kmeans([X';Y'],N); % find centers
%
% PX = zeros(M,N); PY = zeros(M,N);
%
% for j = 1:N
% R = sqrt((X(1,:)-C(j,1)).^2+(X(2,:)-C(j,2)).^2);
% PX(:,j) = PHI(R(:)/d);
% R = sqrt((Y(1,:)-C(j,1)).^2+(Y(2,:)-C(j,2)).^2);
% PY(:,j) = PHI(R(:)/d);
% end
% %% EDMD
% K=PX(1:M,:)\PY(1:M,:);
% [V,LAM]=eig(K,'vector');
%
% %% ResDMD for pseudospectrum, code available here: https://github.com/MColbrook/Residual-Dynamic-Mode-Decomposition
%
% z_pts=kron(x_pts,ones(length(y_pts),1))+1i*kron(ones(1,length(x_pts)),y_pts(:)); z_pts=z_pts(:);
%
% RES = KoopPseudoSpecQR(PX,PY,1/M,z_pts,'Parallel','off');
% RES = reshape(RES,length(y_pts),length(x_pts));
%%
close all
clear;load('duffing3.mat')
figure
contourf(reshape(real(z_pts),length(y_pts),length(x_pts)),reshape(imag(z_pts),length(y_pts),length(x_pts)),log10(max(min(v),real(RES))),log10(v));
hold on
contourf(reshape(real(z_pts),length(y_pts),length(x_pts)),-reshape(imag(z_pts),length(y_pts),length(x_pts)),log10(max(min(v),real(RES))),log10(v));
cbh=colorbar;
cbh.Ticks=log10([0.005,0.01,0.1,1]);
cbh.TickLabels=[0,0.01,0.1,1];
clim([log10(min(v)),0]);
reset(gcf)
set(gca,'YDir','normal')
colormap gray
ax=gca; ax.FontSize=18; axis equal tight; axis([x_pts(1),x_pts(end),-y_pts(end),y_pts(end)])
xlabel('$\mathrm{Re}(\lambda)$','interpreter','latex','fontsize',18)
ylabel('$\mathrm{Im}(\lambda)$','interpreter','latex','fontsize',18)
title(sprintf('$\\mathrm{Sp}_\\epsilon(\\mathcal{K})$, $N=%d$',N),'interpreter','latex','fontsize',18)
exportgraphics(gcf,sprintf('saved_figures/duffing_pseudospec_N%d.png',N),'ContentType','image','BackgroundColor','w','Resolution',200)
figure
plot(cos(0:0.001:2*pi),sin(0:0.001:2*pi),'g','linewidth',2)
hold on
plot(real(LAM),imag(LAM),'.r','markersize',12)
ax=gca; ax.FontSize=18; axis equal tight; axis([x_pts(1),x_pts(end),-y_pts(end),y_pts(end)])
xlabel('$\mathrm{Re}(\lambda)$','interpreter','latex','fontsize',18)
ylabel('$\mathrm{Im}(\lambda)$','interpreter','latex','fontsize',18)
title(sprintf('DMD Eigenvalues, $N=%d$',N),'interpreter','latex','fontsize',18)
exportgraphics(gcf,sprintf('saved_figures/duffing_evals_N%d.pdf',N),'ContentType','vector','BackgroundColor','w')
res = (vecnorm(PY*V-PX*V*diag(LAM))./vecnorm(PX*V))'; % ResDMD for residuals
figure
histogram(res,10,'FaceColor',[0.8500 0.3250 0.0980])
xlabel('$\|(\mathcal{K}-\lambda_j I)g_j\|$','interpreter','latex','fontsize',18)
title(sprintf('Histogram of Errors, $N=%d$',N),'interpreter','latex','fontsize',18)
ax=gca; ax.FontSize=18;
exportgraphics(gcf,sprintf('saved_figures/duffing_hist_N%d.pdf',N),'ContentType','vector','BackgroundColor','w')
%% Plot the eigenfunctions
[~,I]= sort(abs(1-LAM),'ascend');
LAM = LAM(I); V = V(:,I);
close all
xvec = (-2:0.25:0) - 0.01;
xvec = [xvec,-xvec];
for j=1:14
u=real([PX;PY]*V(:,j));
figure1=figure;
axes1 = axes('Parent',figure1);
hold(axes1,'on');
scatter([X(1,:),Y(1,:)],[X(2,:),Y(2,:)],5,u,'filled');
colormap(brighten(brewermap([],'RdYlBu'),-0.3))
clim([min(u),max(u)])
for jj=1:length(xvec)
ODEFUN=@(t,y) [y(2);y(1)-y(1).^3];
[~,Y1]=ode45(ODEFUN,0:0.01:20,[xvec(jj),0],options);
plot(Y1(:,1),Y1(:,2),'k','linewidth',1);
end
axis(axes1,'tight'); grid off; axis off%hold(axes1,'off');
set(axes1,'DataAspectRatio',[1 1 1]);
pause(1)
exportgraphics(gcf,sprintf('saved_figures/duffing_efun_%d.png',j),'ContentType','image','BackgroundColor','w','Resolution',200)
close all
end