This post contains a Matlab example to see how trajectory of the Lorenz attractor is evolving as time flow and the Lorenz attractor is shown below.
Below is the M file used to generate the above picture.
%% Imlements the Lorenz System ODE
% dx/dt = sigma*(y-x)
% dy/dt = x*(rho-z)-y
% dz/dt = xy-beta*z
clear all;
%% Settings for Lorenz system
global sigma rho beta
% Standard constants for the Lorenz Attractor
sigma = 10;
rho = 28;
beta = 8/3;
tmax = 1000;
start_point = [1.0 1.0 1.0];
%% Function declaration
%[t,y] = ode45(@mylorenz,[0 1000], [1.0 1.0 1.0]);
f = @(t,y) [sigma*(y(2) - y(1)); -y(1)*y(3) + rho*y(1) - y(2); y(1)*y(2) - beta*y(3)];
% To plot a trajectory in the phase plane starting at a point (a1, a2) at
% time t=0 for increasing values of t going from 0 to 4 type
[ts,ys] = ode45(f,[0,tmax],start_point);
syms t y1 y2 y3
F = f(t,[y1;y2;y3])
[y1s,y2s,y3s] = solve(F(1),F(2),F(3),y1,y2,y3)
plot3(y1s(1),y2s(1),y3s(1),'*', y1s(2),y2s(2),y3s(2),'>', y1s(3),y2s(3),y3s(3),'o');
strCrit1 = ['(',num2str(double(y1s(1))),',',num2str(double(y2s(1))),',',num2str(double(y3s(1))),')'];
strCrit2 = ['(',num2str(double(y1s(2))),',',num2str(double(y2s(2))),',',num2str(double(y3s(2))),')'];
strCrit3 = ['(',num2str(double(y1s(3))),',',num2str(double(y2s(3))),',',num2str(double(y3s(3))),')'];
%legend(strCrit1,strCrit2,strCrit3,'Location','northoutside','Orientation','horizontal')
legend(strCrit1,strCrit2,strCrit3);
[t,y] = ode45(@mylorenz,[0 tmax], start_point);
h = animatedline;
h.Color = [0 0 1];
%# initialize point
spot = line('XData',y(1,1), 'YData',y(1,2), 'ZData',y(1,3), ...
'Color','r', 'marker','.', 'MarkerSize',50);
for k = 1:length(t)
addpoints(h,y(k,1),y(k,2),y(k,3));
set(spot,'XData',y(k,1), 'YData',y(k,2), 'ZData',y(k,3)) %# update X/Y data
str = ['Time: ',num2str(k)];
%set(hTxt,'String',str); %# update time tick
title(['Time ',str,' Ticks'])
drawnow
if ~ishandle(h), return; end %# end running in case you close the figure
end
grid on;
And another m file: mylorenz.m
% mylorenz.m
%
% Imlements the Lorenz System ODE
% dx/dt = sigma*(y-x)
% dy/dt = x*(rho-z)-y
% dz/dt = xy-beta*z
%
% Inputs:
% t - Time variable: not used here because our equation
% is independent of time, or 'autonomous'.
% x - Independent variable: has 3 dimensions
% Output:
% dx - First derivative: the rate of change of the three dimension
% values over time, given the values of each dimension
function dy = mylorenz(t,y)
global sigma rho beta
% Standard constants for the Lorenz Attractor
%sigma = 10;
%rho = 28;
%beta = 8/3;
% I like to initialize my arrays
dy = [0; 0; 0];
%dy = zeros(3,1);
dy(1) = sigma*(y(2) - y(1));
dy(2) = -y(1)*y(3) + rho*y(1) - y(2);
dy(3) = y(1)*y(2) - beta*y(3);
