A further extension of the original PDE was made, but due to the increased complexity, Maple started to struggle, so I substituted some numerical values.
The LAD procedure within LAD_complex ,still executed the task correctly.
However, continuing further is no longer practical unless the LAD_complex procedure is simplified for more practical use.

Download Uitbreiding_LAD_procedure_naar_complex_uitbreiding_A_12-6-2025mprimes.mw

Download Uitbreiding_LAD_procedure_naar_complex_12-6-2025mprimes.mw

@dharr 
What you could do is approach it more systematically and use the LAD procedure as a basis.
Extend the PDE:

by adding the simplest new term or expression.
A possible check in the form of: 'Extrapolate to an infinite number of terms' probably won't exist for that extended PDE.

Yes, you can choose initial conditions arbitrarily — but not randomly if your goal is to analyze specific behavior.

But Beware:

• Choosing ICs too far from equilibrium can lead to misleading visuals, especially if you're interested in local behavior near a critical point.

• In linear systems, every IC leads to a unique trajectory because the system is deterministic and satisfies the existence and uniqueness theorem.

In this example, it are 4 points of a square around the critical point.

Adding more points and lines,circle 


 

Download complexe_plotdriehoek_en_cirkel10-6-2025mprimes.mw

@jalal 
Hi,
a example to start with 

A := 1 + I;
B := 4 + 2*I;
C := 2 + 5*I;
punt := z -> [Re(z), Im(z)];
driehoek := plot([punt(A), punt(B), punt(C), punt(A)], color = blue, thickness = 2);
punten := pointplot([punt(A), punt(B), punt(C)], symbol = solidcircle, color = red, symbolsize = 15);
display([driehoek, punten], scaling = constrained, title = "Triangle in complex plane", axes = boxed);

It seems that the array notation is not suited for pattern matching in Maple ?

@jalal 
More intuiative with complex numbers ...

corrected code

restart;
with(DEtools);
with(plots);
with(LinearAlgebra);
f := y -> y^2*(1 - y^2);
sys := [diff(x(t), t) = -x(t), diff(y(t), t) = f(y(t))];
trajectories := [[x(0) = 1, y(0) = 1.5], [x(0) = -1, y(0) = 0.5], [x(0) = 0.5, y(0) = -1.5]];
phase_plot := DEplot(sys, [x(t), y(t)], t = 0 .. 10, trajectories, x = -2 .. 2, y = -2 .. 2, arrows = medium, dirfield = [20, 20], linecolor = blue, title = "Phase Portrait with Equilibrium Points");
equilibrium_points := pointplot([[0, 0], [0, 1], [0, -1]], symbol = solidcircle, color = black, symbolsize = 15);
display([phase_plot, equilibrium_points], scaling = constrained);
print("\n### DETAILED STABILITY ANALYSIS ###");
F := [-x, -y^4 + y^2];
J := Matrix([[diff(F[1], x), diff(F[1], y)], [diff(F[2], x), diff(F[2], y)]]);
print("\n1. Analysis for (0,0):");
J0 := eval(J, {x = 0, y = 0});
print("Jacobian:", J0);
print("Eigenvalues:", Eigenvalues(J0));
print("Conclusion:");
print("   - Eigenvalues: -1 and 0");
print("   - Non-hyperbolic point");
print("   - dy/dt ≈ y^2 > 0 for y ≈ 0");
print("   - ⇒ (0,0) is UNSTABLE");
print("\n2. Analysis for (0,1):");
J1 := eval(J, {x = 0, y = 1});
print("Jacobian:", J1);
print("Eigenvalues:", Eigenvalues(J1));
print("Conclusion:");
print("   - Eigenvalues: -1 and -2");
print("   - Both negative ⇒ STABLE NODE");
print("   - All nearby trajectories converge to this point");
print("\n3. Analysis for (0,-1):");
Jm1 := eval(J, {x = 0, y = -1});
print("Jacobian:", Jm1);
print("Eigenvalues:", Eigenvalues(Jm1));
print("Conclusion:");
print("   - Eigenvalues: -1 and +2");
print("   - Mixed eigenvalues ⇒ SADDLE POINT");
print("   - Stable in x-direction, unstable in y-direction");
print("   - ⇒ (0,-1) is UNSTABLE");
print("\n### VISUAL INTERPRETATION ###");
print("1. (0,0): Arrows point away in y-direction - unstable");
print("2. (0,1): All arrows point toward this point - stable");
print("3. (0,-1): Arrows approach in x-direction but diverge in y-direction - saddle point");
print("\n### SYSTEM BEHAVIOR SUMMARY ###");
print("The system exhibits:");
print("- One stable equilibrium at (0,1)");
print("- One unstable equilibrium at (0,0)");
print("- One saddle point at (0,-1)");
print("- Bistability between y=1 and y=-1 states");
print("- All x-values decay to 0 over time");
print("- y-direction determines long-term behavior");

Never trust ai , because node ( 0,-1) is not stable , when i examine it 

@salim-barzani 
You need help from a specialist , i don't know what you are after for.? 

do_loop_met_absoluutfunctie_complex_herschreven7-6-2025mprimes.mw

@salim-barzani 

 

restart;
expr := abs(U[i](x, t))^n*diff(U[i](x, t), x);
expr_rewritten := subs(abs(U[i](x, t))^n = (U[i](x, t)*conjugate(U[i](x, t)))^(n/2), expr);
print(expr_rewritten);

"in here i want to seperate abs(U[i](x,t))^n=U[i](x,t)^n/2*conjugate(U[i](x,t))"   
I am guessing here, no direction where you are after for?

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