Example of Vector Field - Trajectory - Damped Oscillation

 Followings are the code that I wrote in Octave to creates all the plots shown in this page. You may copy these code and play with these codes. Change variables and try yourself until you get your own intuitive understanding.   < Code 1 >   v = -1:0.2:1;   [y1,y2] = meshgrid(v);   u1=y2; u2=-y1 - 0.25 .* y2;     function dy_dt = f(y, t)   dy_dt = zeros (2,1);     dy_dt(1) = y(2);   dy_dt(2) = -y(1) - 0.25 .* y(2); endfunction     hFig = figure(1);   subplot(1,2,1);   hold on; y1_end = 1.0;   for y1_init = -1.0:0.1:y1_end      y0 = [y1_init;1.0];    t = 0:0.2:25.0;    y = lsode ("f", y0, t);      tStr = sprintf("y1(0) = %.2f, y2(0) = %.2f",y0(1),y0(2));      plot(t,y(:,1),'r-',t,y(:,2),'b-');    xlabel('time');    xlim([0 25]);    ylim([-1.5 1.5]);    legend('y2(t)','y1(t)');    %title(tStr);     end;   box on;   hold off;   subplot(1,2,2);   quiver(y1,y2,u1,u2); box on;   hold on; for y1_init = -1.0:0.1:y1_end      y0 = [y1_init;1.0];    t = 0:0.2:25.0;    y = lsode ("f", y0, t);      tStr = sprintf("y1(0) = %.2f, y2(0) = %.2f",y0(1),y0(2));        plot(y(:,1),y(:,2),'r-'); xlabel('y(2)');ylabel('y(1)');    axis([-1 1 -1 1]);      plot(y0(1),y0(2),'bo','MarkerFaceColor',[0 0 1],'MarkerSize',10);    plot(y(end,1),y(end,2),'ro','MarkerFaceColor',[1 0 0],'MarkerSize',10);    %title(tStr);     end;   hold off;     set(hFig,'Position',[300 300 780 300]);