exercise2_2.m

%% Exercise 2

%-------------------------------------------------------------------------%
% Two-dimensional linear systems
%-------------------------------------------------------------------------%

% Date: 09/03/2021
% Author/s: Yi Qiang Ji Zhang
% Subject: Nonlinear Systems, Chaos and Control in Engineering
% Professor: Antonio Pons & Cristina Masoller

% Clear workspace, command window and close windows
clear;
close all;
clc;
% Set interpreter to latex
set(groot,'defaultAxesTickLabelInterpreter','latex');  
set(groot,'defaulttextinterpreter','latex');
set(groot,'defaultLegendInterpreter','latex');

% Function handle
a = 1;
b = -2;
c = 3;
d = 1;


f1 = @(t,x,y) a*x + b*y;
f2 = @(t,x,y) c*x + d*y;


% 1.1 Numerical data
dt = 0.01; % Time steps (dt)
t_final = 10; % time units

for j=1:length(dt)
% N_steps(j) = ceil(t_final/h(j)); % Number of steps, rounds up with ceil
N_steps(j) = length(0:dt:t_final)-1;
end

% Initial conditions
% x0 = -1.25;
% y0 = +1.0;
% x0 = 0.20;
% y0 = +1.5;
x0 = [-1.25 1.0 -1.05];
y0 = [0.2 1.5 +2.5];

% For each h (dt)
for j=1:length(x0)
    t(1) = 0; % We begin at t=0 s
    x(j,1) = x0(j); % x(0)
    y(j,1) = y0(j); % y(0)
        % Euler method's update loop
        for i=1:N_steps
            t(i+1) = t(i) + dt;
            x(j,i+1) = x(j,i) + f1(t(i),x(j,i),y(j,i))*dt;
            y(j,i+1) = y(j,i) + f2(t(i),x(j,i),y(j,i))*dt;
        end
        plot_pdf = figure(1);
        plot(t,x(j,:),t,y(j,:));
        hold on;
        plot_pdf2 = figure(2);
        plot(x(j,:),y(j,:));
        hold on;
end

plot_pdf = figure(1);
xlabel('Time units')
ylabel('Value')
title('\textbf{Two-dimensional linear systems}')
legend('$x_{01} \ \mathbf{c_1}$','$y_{01} \ \mathbf{c_1}$','$x_{02} \ \mathbf{c_2}$', ...
    '$y_{02}  \ \mathbf{c_2}$','$x_{03} \ \mathbf{c_3}$','$y_{03} \ \mathbf{c_3}$', ...
    'location','southwest')
box on
grid minor
hold off;


% Save pdf
set(plot_pdf, 'Units', 'Centimeters');
pos = get(plot_pdf, 'Position');
set(plot_pdf, 'PaperPositionMode', 'Auto', 'PaperUnits', 'Centimeters', ...
    'PaperSize',[pos(3), pos(4)]);
print(plot_pdf, '2DLinearSystem_2_all_init_cond.pdf', '-dpdf', '-r0');

% Save png
print(gcf,'2DLinearSystem_2_all_init_cond.png','-dpng','-r600');


plot_pdf2 = figure(2);
xlabel('x')
ylabel('y')
title('\textbf{Two-dimensional linear systems}')
legend('$\mathbf{c_1}$','$\mathbf{c_2}$', '$\mathbf{c_3}$', ...
    'location','best')
box on
grid minor

% Save pdf
set(plot_pdf2, 'Units', 'Centimeters');
pos = get(plot_pdf2, 'Position');
set(plot_pdf2, 'PaperPositionMode', 'Auto', 'PaperUnits', 'Centimeters', ...
    'PaperSize',[pos(3), pos(4)]);
print(plot_pdf2, '2DLinearSystem_all_init_cond_xy.pdf', '-dpdf', '-r0');

% Save png
print(gcf,'2DLinearSystem_all_init_cond_xy.png','-dpng','-r600');