torabi

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These are questions asked by torabi

Can anyone  produce these diagram?  Please Read the theory in:

   https://en.wikipedia.org/wiki/Logistic_map.

Wikipedia pages that explain bifurcation diagrams and attractors in more elementary contexts.

See the bifurcation diagram in the picture

   https://en.wikipedia.org/wiki/Logistic_map#/media/File:Logistic_Bifurcation_map_High_Resolution.png

 

How I can convert Root Of to conventional form?

Thanks

root_of.mw
 

Q1 := x*(x-delta)*(1-x)*(x+y)-alpha*x*y; Q2 := beta*x*y-Zeta*y*(x+y); SOLL := solve({Q1, Q2}, {x, y})

{x = 0, y = 0}, {x = 1, y = 0}, {x = delta, y = 0}, {x = RootOf(beta*_Z^2+(-beta*delta-beta)*_Z-alpha*Zeta+alpha*beta+delta*beta), y = -RootOf(beta*_Z^2+(-beta*delta-beta)*_Z-alpha*Zeta+alpha*beta+delta*beta)*(Zeta-beta)/Zeta}

(1)

``


 

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how I can remove this error.

Error, numeric exception: division by zero

code2.mw

result3.mw
 

NULL

with(LinearAlgebra); restart; delta := 0.407e-9; C := 2023.2; L := 5*delta

0.2035e-8

(1)

eq := (1/12)*delta^2*S^4+S^2+omega^2/C^2

0.1380408333e-19*S^4+S^2+0.2442993814e-6*omega^2

(2)

solve(eq, S)

0.7244233290e-6*(-0.6902041665e32+2760816666.*(-0.8430822546e19*omega^2+0.6250000000e45)^(1/2))^(1/2), -0.7244233290e-6*(-0.6902041665e32+2760816666.*(-0.8430822546e19*omega^2+0.6250000000e45)^(1/2))^(1/2), 0.7244233290e-6*(-0.6902041665e32-2760816666.*(-0.8430822546e19*omega^2+0.6250000000e45)^(1/2))^(1/2), -0.7244233290e-6*(-0.6902041665e32-2760816666.*(-0.8430822546e19*omega^2+0.6250000000e45)^(1/2))^(1/2)

(3)

U := _C1*exp(sqrt(-2*C*(3*C-sqrt(-3*delta^2*omega^2+9*C^2)))*x/(C*delta))+_C2*exp(-sqrt(-2*C*(3*C-sqrt(-3*delta^2*omega^2+9*C^2)))*x/(C*delta))+_C3*exp(sqrt(-2*C*(3*C+sqrt(-3*delta^2*omega^2+9*C^2)))*x/(C*delta))+_C4*exp(-sqrt(-2*C*(3*C+sqrt(-3*delta^2*omega^2+9*C^2)))*x/(C*delta))

U2 := diff(U, x)

1214414.026*_C1*(-24560029.44+4046.4*(-0.496947e-18*omega^2+36840044.16)^(1/2))^(1/2)*exp(1214414.026*(-24560029.44+4046.4*(-0.496947e-18*omega^2+36840044.16)^(1/2))^(1/2)*x)-1214414.026*_C2*(-24560029.44+4046.4*(-0.496947e-18*omega^2+36840044.16)^(1/2))^(1/2)*exp(-1214414.026*(-24560029.44+4046.4*(-0.496947e-18*omega^2+36840044.16)^(1/2))^(1/2)*x)+1214414.026*_C3*(-24560029.44-4046.4*(-0.496947e-18*omega^2+36840044.16)^(1/2))^(1/2)*exp(1214414.026*(-24560029.44-4046.4*(-0.496947e-18*omega^2+36840044.16)^(1/2))^(1/2)*x)-1214414.026*_C4*(-24560029.44-4046.4*(-0.496947e-18*omega^2+36840044.16)^(1/2))^(1/2)*exp(-1214414.026*(-24560029.44-4046.4*(-0.496947e-18*omega^2+36840044.16)^(1/2))^(1/2)*x)

(4)

A := simplify(Matrix(4, 4, [[coeff(subs(x = 0, U), _C1), coeff(subs(x = 0, U), _C2), coeff(subs(x = 0, U), _C3), coeff(subs(x = 0, U), _C4)], [coeff(subs(x = L, U), _C1), coeff(subs(x = L, U), _C2), coeff(subs(x = L, U), _C3), coeff(subs(x = L, U), _C4)], [coeff(subs(x = L, U2), _C1), coeff(subs(x = L, U2), _C2), coeff(subs(x = L, U2), _C3), coeff(subs(x = L, U2), _C4)], [coeff(subs(x = 0, U2), _C1), coeff(subs(x = 0, U2), _C2), coeff(subs(x = 0, U2), _C3), coeff(subs(x = 0, U2), _C4)]]))

Matrix(%id = 18446746958277761134)

(5)

solve(Determinant(A) = 0)

Warning,  computation interrupted

 

NULL

NULL

``


 

Download result3.mw
 

NULL

with(LinearAlgebra); restart; delta := 0.407e-9; C := 2023.2; L := 5*delta

0.2035e-8

(1)

eq := (1/12)*delta^2*S^4+S^2+omega^2/C^2

0.1380408333e-19*S^4+S^2+0.2442993814e-6*omega^2

(2)

solve(eq, S)

0.7244233290e-6*(-0.6902041665e32+2760816666.*(-0.8430822546e19*omega^2+0.6250000000e45)^(1/2))^(1/2), -0.7244233290e-6*(-0.6902041665e32+2760816666.*(-0.8430822546e19*omega^2+0.6250000000e45)^(1/2))^(1/2), 0.7244233290e-6*(-0.6902041665e32-2760816666.*(-0.8430822546e19*omega^2+0.6250000000e45)^(1/2))^(1/2), -0.7244233290e-6*(-0.6902041665e32-2760816666.*(-0.8430822546e19*omega^2+0.6250000000e45)^(1/2))^(1/2)

(3)

U := _C1*exp(sqrt(-2*C*(3*C-sqrt(-3*delta^2*omega^2+9*C^2)))*x/(C*delta))+_C2*exp(-sqrt(-2*C*(3*C-sqrt(-3*delta^2*omega^2+9*C^2)))*x/(C*delta))+_C3*exp(sqrt(-2*C*(3*C+sqrt(-3*delta^2*omega^2+9*C^2)))*x/(C*delta))+_C4*exp(-sqrt(-2*C*(3*C+sqrt(-3*delta^2*omega^2+9*C^2)))*x/(C*delta))

U2 := diff(U, x)

1214414.026*_C1*(-24560029.44+4046.4*(-0.496947e-18*omega^2+36840044.16)^(1/2))^(1/2)*exp(1214414.026*(-24560029.44+4046.4*(-0.496947e-18*omega^2+36840044.16)^(1/2))^(1/2)*x)-1214414.026*_C2*(-24560029.44+4046.4*(-0.496947e-18*omega^2+36840044.16)^(1/2))^(1/2)*exp(-1214414.026*(-24560029.44+4046.4*(-0.496947e-18*omega^2+36840044.16)^(1/2))^(1/2)*x)+1214414.026*_C3*(-24560029.44-4046.4*(-0.496947e-18*omega^2+36840044.16)^(1/2))^(1/2)*exp(1214414.026*(-24560029.44-4046.4*(-0.496947e-18*omega^2+36840044.16)^(1/2))^(1/2)*x)-1214414.026*_C4*(-24560029.44-4046.4*(-0.496947e-18*omega^2+36840044.16)^(1/2))^(1/2)*exp(-1214414.026*(-24560029.44-4046.4*(-0.496947e-18*omega^2+36840044.16)^(1/2))^(1/2)*x)

(4)

A := simplify(Matrix(4, 4, [[coeff(subs(x = 0, U), _C1), coeff(subs(x = 0, U), _C2), coeff(subs(x = 0, U), _C3), coeff(subs(x = 0, U), _C4)], [coeff(subs(x = L, U), _C1), coeff(subs(x = L, U), _C2), coeff(subs(x = L, U), _C3), coeff(subs(x = L, U), _C4)], [coeff(subs(x = L, U2), _C1), coeff(subs(x = L, U2), _C2), coeff(subs(x = L, U2), _C3), coeff(subs(x = L, U2), _C4)], [coeff(subs(x = 0, U2), _C1), coeff(subs(x = 0, U2), _C2), coeff(subs(x = 0, U2), _C3), coeff(subs(x = 0, U2), _C4)]]))

Matrix(%id = 18446746958277761134)

(5)

solve(Determinant(A) = 0)

Warning,  computation interrupted

 

NULL

NULL

``


 

Download result3.mw

 

 

how i can write these boundary conditions for dsolve?

(diff(u(r), r))^(n-1)*(diff(r*(diff(u(r), r)), r))/r    be [finite] at r =0

and  u(0) = finite?

Thanks..

 

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