ldedemos.html

> restart;

> with(LinearAlgebra): with(DEtools): with(plots):

> buildsys := proc(mat)

> {diff(x(t),t)=mat[1,1]*x(t)+mat[1,2]*y(t), diff(y(t),t)=mat[2,1]*x(t)+mat[2,2]*y(t)};

> end;

> circlepts := proc(numints)
dt := 2*Pi/numints; plist :=[];
for i from 1 to numints do t := (i-1)*dt; plist:=[op(plist), <cos(t),sin(t)>]; od;
return Matrix(plist);
end:

Warning, `dt` is implicitly declared local to procedure `circlepts`

Warning, `plist` is implicitly declared local to procedure `circlepts`

Warning, `i` is implicitly declared local to procedure `circlepts`

Warning, `t` is implicitly declared local to procedure `circlepts`

> cpts := circlepts(12);

Typesetting:-mprintslash([`:=`(cpts, RTABLE(8700000, anything, Matrix, rectangular, Fortran_order, [], 2, 1 .. 2, 1 .. 12))], [Matrix(%id = 8700000)])

> mkinitials := proc(ptmat)
[seq([0, ptmat[1,i], ptmat[2,i]], i=1..ColumnDimension(ptmat))];
end;

> mkinitials(cpts);

> P := <<1,-2>|<1,1>>;

> dmat := DiagonalMatrix([-1/5, -2/5]);

Typesetting:-mprintslash([`:=`(dmat, Matrix([[-`/`(1, 5), 0], [0, -`/`(2, 5)]]))], [Matrix(%id = 9301872)])

> eqns := buildsys(dmat);

> DEplot(buildsys(dmat), [x(t), y(t)], t=0..10, mkinitials(cpts), animatecurves=true, color=magnitude, linecolor=blue);

Plot_2d

> A := P.dmat.P^(-1);

Typesetting:-mprintslash([`:=`(A, Matrix([[-`/`(1, 3), -`/`(1, 15)], [-`/`(2, 15), -`/`(4, 15)]]))], [Matrix(%id = 10399544)])

> pcpts := P.cpts

Warning, inserted missing semicolon at end of statement

Typesetting:-mprintslash([`:=`(pcpts, RTABLE(10236304, anything, Matrix, rectangular, Fortran_order, [], 2, 1 .. 2, 1 .. 12))], [Matrix(%id = 10236304)])

> DEplot(buildsys(A), [x(t), y(t)], t=0..10, mkinitials(pcpts), animatecurves=true, color=magnitude, linecolor=blue);

Plot_2d

> dmat := DiagonalMatrix([0.1,0.15]);

> DEplot(buildsys(dmat), [x(t), y(t)], t=0..15, mkinitials(cpts), animatecurves=true, color=magnitude, linecolor=blue);

Plot_2d

> A := P.dmat.P^(-1);

> pcpts := P.cpts;

Typesetting:-mprintslash([`:=`(pcpts, RTABLE(11299192, anything, Matrix, rectangular, Fortran_order, [], 2, 1 .. 2, 1 .. 12))], [Matrix(%id = 11299192)])

> DEplot(buildsys(A), [x(t), y(t)], t=0..15, mkinitials(pcpts), animatecurves=true, color=magnitude, linecolor=blue);

Plot_2d

> pf := DEplot(buildsys(A), [x(t), y(t)], t=0..15, mkinitials(pcpts), color=magnitude, linecolor=blue):

> p1 := plot([P[1,1]*t, P[2,1]*t, t=-7..7], color=green):

> p2 := plot([P[1,2]*t, P[2,2]*t, t=-7..7], color=red):

> display([pf,p1,p2]);

Plot_2d

> dmat := DiagonalMatrix([-0.1,0.04]);

> DEplot(buildsys(dmat), [x(t), y(t)], t=0..15, mkinitials(circlepts(16)), x=-2..2, y=-2..2, animatecurves=true, color=magnitude, linecolor=blue);

Plot_2d

> A := P.dmat.P^(-1);

> ppts := P.circlepts(16);

Typesetting:-mprintslash([`:=`(ppts, RTABLE(16019296, anything, Matrix, rectangular, Fortran_order, [], 2, 1 .. 2, 1 .. 16))], [Matrix(%id = 16019296)])

> DEplot(buildsys(A), [x(t), y(t)], t=0..15, mkinitials(ppts), x=-2..2, y=-2..2, animatecurves=true, color=magnitude, linecolor=blue);

Plot_2d

> alpha :=1 ; beta := 2; lambda := alpha + beta*I;

> Q := copy(P);

> M := <<alpha|-beta>, <beta|alpha>>;

> A := Q.M.Q^(-1);

Typesetting:-mprintslash([`:=`(A, Matrix([[`/`(1, 3), -`/`(4, 3)], [`/`(10, 3), `/`(5, 3)]]))], [Matrix(%id = 12857064)])

> Eigenvectors(A);

Typesetting:-mprintslash([Vector[column]([[`+`(1, `*`(2, `*`(I)))], [`+`(1, `-`(`*`(2, `*`(I))))]]), Matrix([[`+`(`-`(`/`(1, 5)), `*`(`/`(3, 5), `*`(I))), `+`(`-`(`/`(1, 5)), `-`(`*`(`/`(3, 5), `*`(I)...

> (u,v):=Column(Q, 1..-1);

> P:= < u-I*v|u+I*v>;

> P^(-1).A.P;

> alpha := 0.1; beta := 2;

> M := Matrix([

> [alpha, -beta],

> [beta, alpha]]);

> DEplot(buildsys(M), [x(t),y(t)], t=0..5, [[0, 1, 0]], animatecurves=true, color=magnitude, linecolor=blue,numsteps=1000, numpoints = 505);

Plot_2d

> A := Q.M.Q^(-1);

> apt := A.<1,0>;

> DEplot(buildsys(A), [x(t),y(t)], t=0..5, [[0,apt[1], apt[2]]], animatecurves=true, color=magnitude, linecolor=blue,numsteps=1000, numpoints = 505);

Plot_2d

> alpha := -0.1; beta := 2;

> M := Matrix([

> [alpha, -beta],

> [beta, alpha]]);

> DEplot(buildsys(M), [x(t),y(t)], t=0..7, [[0, 1, 0]], animatecurves=true, color=magnitude, linecolor=blue,numsteps=1000, numpoints = 505);

Plot_2d

> A := Q.M.Q^(-1);

> apt := A.<1,0>;

> DEplot(buildsys(A), [x(t),y(t)], t=0..7, [[0,apt[1], apt[2]]], animatecurves=true, color=magnitude, linecolor=blue,numsteps=1000, numpoints = 505);

Plot_2d

> alpha := 0; beta := 2;

> M := Matrix([

> [alpha, -beta],

> [beta, alpha]]);

> DEplot(buildsys(M), [x(t),y(t)], t=0..4, x=-1.1..1.1, y=-1.1..1.1, [[0, 1, 0]], animatecurves=true, color=magnitude, linecolor=blue,numsteps=1000, numpoints = 505);

Plot_2d

> A := Q.M.Q^(-1);

Typesetting:-mprintslash([`:=`(A, Matrix([[-`/`(2, 3), -`/`(4, 3)], [`/`(10, 3), `/`(2, 3)]]))], [Matrix(%id = 11326016)])

> apt := A.<1,0>;

Typesetting:-mprintslash([`:=`(apt, Vector[column]([[-`/`(2, 3)], [`/`(10, 3)]]))], [Vector[column](%id = 7962344)])

> DEplot(buildsys(A), [x(t),y(t)], t=0..7, [[0,apt[1], apt[2]]],x=-4..4, y=-4..4, animatecurves=true, color=magnitude, linecolor=blue,numsteps=1000, numpoints = 505);

Plot_2d

> lambda := -0.1;

> M := Matrix([

> [lambda, 1],

> [0, lambda]]);

> DEplot(buildsys(M), [x(t), y(t)], t=0..30, mkinitials(cpts), animatecurves=true, color=magnitude, linecolor=blue);

Plot_2d

> sol := dsolve(buildsys(M), [x(t),y(t)]);

$Typesetting:-mprintslash([`:=`(sol, {x(t) = `*`(`+`(`*`(_C2, `*`(t)), _C1), `*`(exp(`+`(`-`(`*`(`/`(1, 10), `*`(t))))))), y(t) = `*`(_C2, `*`(exp(`+`(`-`(`*`(`/`(1, 10), `*`(t)))))))})], [{x(t) = `*`(...$

> subs(sol, y(t)/x(t));

> limit(%, t=infinity);

> P := copy(Q);

> A := P.M.P^(-1);

> apts := P.cpts;

Typesetting:-mprintslash([`:=`(apts, RTABLE(14159784, anything, Matrix, rectangular, Fortran_order, [], 2, 1 .. 2, 1 .. 12))], [Matrix(%id = 14159784)])

> DEplot(buildsys(A), [x(t), y(t)], t=0..30, mkinitials(apts), animatecurves=true, color=magnitude, linecolor=blue);

Plot_2d

> lambda := 0.2;

> M := Matrix([

> [lambda, 1],

> [0, lambda]]);

> ippts := Matrix([seq(<i*0.1,-i*0.1/10>, i=-10..10)]);

Typesetting:-mprintslash([`:=`(ippts, RTABLE(23915152, anything, Matrix, rectangular, Fortran_order, [], 2, 1 .. 2, 1 .. 21))], [Matrix(%id = 23915152)])

> DEplot(buildsys(M), [x(t), y(t)], t=0..10, mkinitials(ippts), animatecurves=true, color=magnitude, linecolor=blue);

Plot_2d

> A := P.M.P^(-1);

> apts := P.ippts;

Typesetting:-mprintslash([`:=`(apts, RTABLE(7866880, float[8], Matrix, rectangular, Fortran_order, [], 2, 1 .. 2, 1 .. 21))], [Matrix(%id = 7866880)])

> DEplot(buildsys(A), [x(t), y(t)], t=0..10, mkinitials(apts), animatecurves=true, color=magnitude, linecolor=blue);

Plot_2d

> lambda := 0;

> M := Matrix([

> [lambda, 1],

> [0, lambda]]);

> ippts := Matrix([seq(<i*0.1,-i*0.1/10>, i=-10..10)]);

Typesetting:-mprintslash([`:=`(ippts, RTABLE(8412328, anything, Matrix, rectangular, Fortran_order, [], 2, 1 .. 2, 1 .. 21))], [Matrix(%id = 8412328)])

> DEplot(buildsys(M), [x(t), y(t)], t=0..10, mkinitials(ippts), animatecurves=true, color=magnitude, linecolor=blue);

Plot_2d

> dsolve(buildsys(M), [x(t),y(t)]);

${y(t) = _C2, x(t) = `+`(`*`(_C2, `*`(t)), _C1)}$

> A := P.M.P^(-1);

Typesetting:-mprintslash([`:=`(A, Matrix([[`/`(2, 3), `/`(1, 3)], [-`/`(4, 3), -`/`(2, 3)]]))], [Matrix(%id = 16132656)])

> apts := P.ippts;

Typesetting:-mprintslash([`:=`(apts, RTABLE(9295840, float[8], Matrix, rectangular, Fortran_order, [], 2, 1 .. 2, 1 .. 21))], [Matrix(%id = 9295840)])