Example_Lorenz.m

%Reconstruction of Lorenz system using ABM algorithm and delMinorTerms
rng default;
close all;

%simulate Lorenz system
Tmax = 45;
h = 0.01;
[t,y] = ode45(@Lorenz,[0:h:Tmax],[0.1,0,-0.1]); %solve ODE
w = transpose(Lorenz(0,transpose(y))); %find derivatives

%plot x-z plane projection
figure(1);
plot3(y(:,1),y(:,2),y(:,3));
xlabel('\itx');
ylabel('\ity');
zlabel('\itz');

%get uniformly distributed points from the simulated attractor
N = 19; %data points
M = 3; %dim
[Ns, ~] = size(y);
W = zeros(N,M);
Y = zeros(N,M);

for i = 1:N %take random points from attractor
    id = ceil(rand*Ns);  %number of data point
    W(i,:) = w(id,:); %X
    Y(i,:) = y(id,:); %Y
end

%plot sample points
figure(1); hold on
scatter3(Y(:,1),Y(:,2),Y(:,3),23,'MarkerEdgeColor','g','MarkerFaceColor','y','LineWidth',1.5);

%reconstruct order ideal
eps = 1e-5;

sigma = deglexord(0,2,3);
[~, O] = ApproxBM(Y, eps, sigma) %use approximate Buchberger-Moller algorithm

%Use LSM for fitting the equations with the proper coefficients
eta = 1e-7;
H = cell(1,3);
T = cell(1,3);
%reconstruct each equation
for i = 1:3
    V = W(:,i);
    [hi,tau] = delMinorTerms(Y,V,O,eta); %get equation and basis
    V0 = EvalPoly(hi,Y,tau); 
    norm(V - V0) %check if norm is appropriate
    
    H{1,i} = hi;
    T{1,i} = tau;
end

%simulate results
[~,y] = ode45(@(t,x)oderecon(H,T,t,x),[0:h:Tmax],[0.1,0,-0.1]); %solve ODE
figure(1);
plot3(y(:,1),y(:,2),y(:,3),'-');

%display equations
prettyABM(H,T)