@Alfred_F 

There are some corrections  to make for the procedure FastPursuite 
Handling the ode system in Maple for this procedure can be different on two ways.

As you can see the goat is escaping the wolf ..lol,  message (  try  correct this  and add  coordinat axes)



The procedure approach:

The golden ratio ..



cikels-hexacon_gulden_snede_aklfred_mprimes_12-9-2025.mw

How_plot_and_visualization_2D_and_3D_of_imaginary_part-this_idea_mprimes9-9-2025.mw 

@Alfred_F 
A bijective function cannot be periodic ?

@Alfred_F 
alfred_bolpuzzle_mprimes_14-8-2025.mw

linear__selectie_in_pde_parts_mprimes_11-8-2025.mw

@Andiguys 

nusselt_nummer_mprimes_Question_How_Can_I_Generate_and_Verify_the_results_4-8-2025_-versieA.mw

sol := dsolve([de1, de2, bc], numeric, method = bvp[midrich], approxsoln = [y(x) = 1 - exp(-x), z(x) = exp(-x)], abserr = 0.1e-5, maxmesh = 128)

` Solve using BVP with midrich method (collocation-type)`

Angular Motion Over Time

restart;
grid := proc(M::Matrix) local i, j, Ms, m, n, wks; m, n := op(1, M); Ms := map(convert, M, string); wks := XMLTools:-ToString(_XML_Worksheet(DocumentTools:-Layout:-Table(':-alignment' = ':-center', ':-width' = 20, seq(DocumentTools:-Layout:-Column(':-weight' = 3 + max(map(length, Ms[() .. (), j]))), j = 1 .. n), seq(DocumentTools:-Layout:-Row(seq(DocumentTools:-Layout:-Cell(`_XML_Text-field`("alignment" = "centred", "style" = "Text", Ms[i, j])), j = 1 .. n)), i = 1 .. m)))); streamcall(INTERFACE_TASKTEMPLATE(':-insertdirect', ':-content' = wks)); NULL; end proc;
data := [["Case", "Optimal Pc"], ["μ₁ = 0, μ₂ = 0", "Pc₁"], ["μ₁ = 0, μ₂ > 0", "Pc₂"], ["μ₁ > 0, μ₂ = 0", "Pc₃"]];
grid(Matrix(data));

Is this correct ?


 

Download perbiutation_pde_mprimes_21-7-2025.mw

@Alfred_F 
prove by induction ....

@Math-dashti 

restart;
with(DEtools);
with(plots);
f := (x, y) -> 2*(x - 1)*sin(Pi*y) + (x - 1)^2;
g := (x, y) -> y^2 - y;
lin_sys1 := [diff(u(t), t) = 0, diff(v(t), t) = -v(t)];
lin_sys2 := [diff(u(t), t) = 0, diff(v(t), t) = v(t)];
orig_sys := [diff(x(t), t) = f(x(t), y(t)), diff(y(t), t) = g(x(t), y(t))];
plot_orig1 := DEplot(orig_sys, [x(t), y(t)], t = 0 .. 5, x = 0.5 .. 1.5, y = -0.5 .. 0.5, arrows = SLIM, dirfield = [20, 20], title = "Nonlinear system near (1, 0)", axes = boxed);
plot_orig2 := DEplot(orig_sys, [x(t), y(t)], t = 0 .. 5, x = 0.5 .. 1.5, y = 0.5 .. 1.5, arrows = SLIM, dirfield = [20, 20], title = "Nonlinear system near (1, 1)", axes = boxed);
plot_lin1 := DEplot(lin_sys1, [u(t), v(t)], t = 0 .. 5, u = -0.5 .. 0.5, v = -0.5 .. 0.5, arrows = SLIM, dirfield = [20, 20], title = "Linearized system at (1, 0) → origin (u,v)", axes = boxed);
plot_lin2 := DEplot(lin_sys2, [u(t), v(t)], t = 0 .. 5, u = -0.5 .. 0.5, v = -0.5 .. 0.5, arrows = SLIM, dirfield = [20, 20], title = "Linearized system at (1, 1) → origin (u,v)", axes = boxed);
display([plot_orig1, plot_lin1, plot_orig2, plot_lin2], title = "Comparison: Nonlinear vs. Linearized Systems at Critical Points", insequence = false, layout = [[1, 2], [3, 4]]);


with(LinearAlgebra);
with(DEtools);
with(plots);
fx := (x, y) -> 2*(x - 1)*sin(Pi*y) + (x - 1)^2;
fy := (x, y) -> y^2 - y;
solve({fx(x, y) = 0, fy(x, y) = 0}, {x, y});
J := Matrix(2, 2, [diff(fx(x, y), x), diff(fx(x, y), y), diff(fy(x, y), x), diff(fy(x, y), y)]);
J10 := eval(J, [x = 1, y = 0]);
J11 := eval(J, [x = 1, y = 1]);
print("Jacobiaan bij (1,0):", J10);
print("Jacobiaan bij (1,1):", J11);
sys_uv_10 := {diff(u(t), t) = Pi*v(t), diff(v(t), t) = -v(t)};
sys_uv_11 := {diff(u(t), t) = -Pi*v(t), diff(v(t), t) = v(t)};
field_orig_10 := DEplot([diff(x(t), t) = fx(x(t), y(t)), diff(y(t), t) = fy(x(t), y(t))], [x(t), y(t)], t = 0 .. 5, x = 0.5 .. 1.5, y = -0.5 .. 0.5, dirfield = [20, 20], arrows = SLIM, title = "Niet-lineair systeem rond (1,0)");
field_orig_11 := DEplot([diff(x(t), t) = fx(x(t), y(t)), diff(y(t), t) = fy(x(t), y(t))], [x(t), y(t)], t = 0 .. 5, x = 0.5 .. 1.5, y = 0.5 .. 1.5, dirfield = [20, 20], arrows = SLIM, title = "Niet-lineair systeem rond (1,1)");
field_lin_10 := DEplot([op(sys_uv_10)], [u(t), v(t)], t = 0 .. 5, u = -0.5 .. 0.5, v = -0.5 .. 0.5, dirfield = [20, 20], arrows = SLIM, title = "Linearisatie rond (1,0): u' = πv, v' = -v");
field_lin_11 := DEplot([op(sys_uv_11)], [u(t), v(t)], t = 0 .. 5, u = -0.5 .. 0.5, v = -0.5 .. 0.5, dirfield = [20, 20], arrows = SLIM, title = "Linearisatie rond (1,1): u' = -πv, v' = v");
display([field_orig_10, field_lin_10, field_orig_11, field_lin_11], title = "Origineel versus lineair systeem rond kritieke punten");