#
# Phase portraits for systems of linear DEs (given by their matrix) 
# with eigenvalues of the form: 
#     imaginary w/ Real part negative
#     repeated positive
#     distinct, real, positive
#     real, negative
#     real, 1 positive, 1 negative
#
> restart;with(DEtools):with(linalg):with(plots):
Warning: new definition for   norm
Warning: new definition for   trace

# Here, the eigenvalues are negative imaginary, so we get a spiral 
# sink.
> A := linalg[matrix](2,2,[a,b,c,d]):
> a := 0: b := -2: c := 1: d := -2: 
> eigenvals(A); eigenvects(A,radical);

                                - 1 + I, - 1 - I

     [- 1 + I, 1, {[ 1, 1/2 - 1/2 I ]}], [- 1 - I, 1, {[ 1, 1/2 + 1/2 I ]}]
> 
> phaseportrait(A,[x,y],-4..1,{[0,1,0],[0,0,1],[0,0,-1],[0,-1,0],[0,-5,5],[0,5,-5]},stepsize=.2,x=-10..10,y=-10..10,title=`spiral sink`,arrows=THICK);
> exponential(A*t);

    [ exp(- t) cos(t) + exp(- t) sin(t)         - 2 exp(- t) sin(t)        ]
    [                                                                      ]
    [          exp(- t) sin(t)           exp(- t) cos(t) - exp(- t) sin(t) ]
# Here, the eigenvalues are positive repeated, so everything 
# comes along one vector.
>  B := linalg[matrix](2,2,[e,f,g,h]):\
 e := 4: f:= 6: g := 0: h := 4:\
phaseportrait(B,[x,y],-4..1,{[0,1,0],[0,0,1],[0,0,-1],[0,-1,0],[0,-5,5],[0,5,-5],[0,-1,8],[0,1,-8]},stepsize=.2,x=-10..10,y=-10..10,title=`repeated root`,arrows=THICK);\
\

> eigenvals(B); eigenvects(B,radical);\


                                      4, 4

                               [4, 2, {[ 1, 0 ]}]
# Here, the eigenvalues are real and positive (unstable node)
> exponential(B*t);
> J:=jordan(B,P);
> Q:=evalm(1/P);

                                       [ 6  1 ]
                                  Q := [      ]
                                       [ 0  1 ]

                                       [ 4  1 ]
                                  J := [      ]
                                       [ 0  4 ]

                           [ exp(4 t)  6 t exp(4 t) ]
                           [                        ]
                           [     0       exp(4 t)   ]
> A1:= linalg[matrix](2,2,[a1,b1,c1,d1]):
> a1 := -1: b1 := 2: c1 := -7: d1 := 8: 
> eigenvals(A1);eigenvects(A1,radical);

                                      6, 1

                    [6, 1, {[ 1, 7/2 ]}], [1, 1, {[ 1, 1 ]}]
> sol:=c3*exp(6*t)*matrix(2,1,[1,1])+c2*exp(1*t)*matrix(2,1,[1,7/2]);

                                     [ 1 ]             [  1  ]
                  sol := c3 exp(6 t) [   ] + c2 exp(t) [     ]
                                     [ 1 ]             [ 7/2 ]
> exponential(A1*t);

          [  7/5 exp(t) - 2/5 exp(6 t)    2/5 exp(6 t) - 2/5 exp(t)  ]
          [                                                          ]
          [ - 7/5 exp(6 t) + 7/5 exp(t)  - 2/5 exp(t) + 7/5 exp(6 t) ]
> c3:=1; c2:=2; sol1:=evalm(sol); 

                                     c3 := 1

                                     c2 := 2

                                 [ exp(6 t) + 2 exp(t) ]
                         sol1 := [                     ]
                                 [ exp(6 t) + 7 exp(t) ]
> x1:=sol1[1,1]; x2:=sol1[2,1];

                            x1 := exp(6 t) + 2 exp(t)

                            x2 := exp(6 t) + 7 exp(t)
>  plot(x1, t=0..5);
>  plot(x2, t=0..5);
> phaseportrait(A1,[x,y],-4..1,{[0,1,0],[0,0,1],[0,0,-1],[0,-1,0],[0,-5,5],[0,5,-5]},stepsize=.2,x=-10..10,y=-10..10,title=`improper node`,arrows=THICK);
# Here, the eigenvalues are real and negative (stable node)
> 
> restart; with(DEtools):with(linalg):
Error, too many files simultaneously open for reading
Warning: new definition for   norm
Warning: new definition for   trace

> A := linalg[matrix](2,2,[a,b,c,d]):
> 
> a := -3: b := 1: c := 0: d := -2:
> eigenvals(A);eigenvects(A,radical);

                                     -3, -2

                    [-3, 1, {[ 1, 0 ]}], [-2, 1, {[ 1, 1 ]}]
> phaseportrait(A,[x,y],-4..1,{[0,1,0],[0,0,1],[0,0,-1],[0,-1,0],[0,-5,5],[0,5,-5]},stepsize=.2,x=-10..10,y=-10..10,title=`improper node`,arrows=THICK);
# Here, the eigenvalues are one pos, one neg (an unstable saddle)
# 
> 
> A3 := linalg[matrix](2,2,[a3,b3,c3,d3]):
> a3 := 1: b3 := 2: c3 := 3: d3 := 2: 
> eigenvals(A3);eigenvects(A,radical);

                                      4, -1

                    [-3, 1, {[ 1, 0 ]}], [-2, 1, {[ 1, 1 ]}]
> phaseportrait(A3,[x,y],-4..1,{[0,1,0],[0,0,1],[0,0,-1],[0,-1,0],[0,-5,5],[0,5,-5]},stepsize=.2,x=-10..10,y=-10..10,title=`saddle`,arrows=THICK);