% This code is the supplimentary to the problem set 2. 
% It demonstrates the use of the function streamsclice and quiver to plot 
% phase portraits and vector fields in 2D. 
%
% Your task is to modify the code to use in various problems in pset 2. 


% Establish the system to be plotted

x = -4:.01:4;  % x is in the range from -4 to 4
y = -4:.01:4;  % y is in the same range 
[X Y] = meshgrid(x,y);  
   % meshgrid transforms the domain specified by vectors
   % x and y into arrays X and Y that can be used for the evaluation
   % of functions of two variables and 3-D surface plots.
   % The rows of the output array X are copies of the vector x and
   % the columns of the output array Y are copies of the vector y.
                        
% Establish the vector-field, in this example it is x' = xy-1, y' = x-y^3.
% The dots in vector multiplication X.*Y mean that the multiplication is
% done component by component
Xdot = X.*Y-1;
Ydot = X - Y.^3;

% Plot the phase portriat in figure(1) in the first subplot
figure(1); 
subplot(1,2,1); % to learn more about subplot type in MATLAB command window
                % help subplot

%^^^^^^^^^^^ the funciton streamslice plots the phase portrait                
streamslice(X,Y,Xdot,Ydot);

hold on
axis equal
grid on 

% Plotting the fixed points.  Use the dot for stable fixed points and the
% circle for unstable points.  Plot in red.
plot(-1,-1,'.r','MarkerSize',18)
plot( 1,1,'or','MarkerSize',6)

% Label the two axes for the system.
xlabel('X')
ylabel('Y')
title('Phase portrait')


%-----------------------------------------------
% in a second subplot plot the vector field only.
% this is done with function quiver. Sometimes it is useful.
subplot(1,2,2); 

x = -2:.25:2;   
y = -2:.25:2;  
[X Y] = meshgrid(x,y);  
   
Xdot = X.*Y-1;
Ydot = X - Y.^3;

% ^^^^^^^^ the only new function here -- quiver -- plots the vector field
%          given by Xdot and Ydot at the points (X,Y)
quiver(X,Y,Xdot,Ydot);

hold on;  axis equal; grid on; 
plot(-1,-1,'.r','MarkerSize',18)
plot( 1,1,'or','MarkerSize',6)
xlabel('X'); ylabel('Y'); title('Vector Field')