#
# Phase portraits for systems of linear DEs (given by matrices) with
# eigenvalues of the form:
#    complex, real part positive
#    pure imaginary
#    real, one is 0 , other is negative
#    
# Written by Denise Kirschner
#
> restart;with(DEtools):with(linalg):with(plots):
Warning: new definition for   norm
Warning: new definition for   trace

# Here, the eigenvalues are Imaginary positive
> 
> A4 := linalg[matrix](2,2,[a4,b4,c4,d4]):
> a4 := 3: b4 := -2: c4 := 4: d4 := -1: 
> eigenvals(A4); eigenvects(A4,radical);

                                1 + 2 I, 1 - 2 I

           [1 + 2 I, 1, {[ 1, 1 - I ]}], [1 - 2 I, 1, {[ 1, 1 + I ]}]
> phaseportrait(A4,[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 source`,arrows=THICK);\

# 
# Here the eigenvalues are purely imaginary (neutrally stable)
> restart; with(DEtools):with(linalg):
Warning: new definition for   norm
Warning: new definition for   trace

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

                                     I, - I

                   [I, 1, {[ 1, I ]}], [- I, 1, {[ 1, - 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=`center node`,arrows=THICK);
# Here, there is one zero eigenvalue, one negative (stable)
> 
> restart; with(DEtools):with(linalg):
Warning: new definition for   norm
Warning: new definition for   trace

> A := linalg[matrix](2,2,[a,b,c,d]):
> a := 0: b := 0: c := 0: d := -5: 
> eigenvals(A); eigenvects(A,radical);

                                      0, -5

                     [0, 1, {[ 1, 0 ]}], [-5, 1, {[ 0, 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=`infinite line of steady states`,arrows=THICK);
# Here is a proper node: (independent eigenvectors for repeated
# roots)
# 
> restart; with(DEtools):with(linalg):
Warning: new definition for   norm
Warning: new definition for   trace

> A := linalg[matrix](3,3,[4,2,3,2,1,2,-1,2,0]);

                                     [  4  2  3 ]
                                     [          ]
                                A := [  2  1  2 ]
                                     [          ]
                                     [ -1  2  0 ]
> eigenvals(A);eigenvects(A,radical);

                                    5, 1, -1

   [1, 1, {[ -1, 0, 1 ]}], [5, 1, {[ 2, 1, 0 ]}], [-1, 1, {[ 1/2, 1, -3/2 ]}]
> A := linalg[matrix](3,3,[1,0,0,0,1,0,0,0,1]);

                                     [ 1  0  0 ]
                                     [         ]
                                A := [ 0  1  0 ]
                                     [         ]
                                     [ 0  0  1 ]
> eigenvals(A);eigenvects(A,radical);

                                     1, 1, 1

                 [1, 3, {[ 1, 0, 0 ], [ 0, 0, 1 ], [ 0, 1, 0 ]}]
>