Advanced Search

Question: Find the jacobian matrix in two different point and plot bihavior ?

Question:Find the jacobian matrix in two different point and plot bihavior ?

Math-dashti 95 Maple 2024
ode homework differential-equations jacobian phaseportrait plotting
June 25 2025
2

Regarding to this equation i want to do the same as paper did, but i am not sure how i can determine the both  point in plot and how thus point show effect of system of equation and  how find jacobian in the two different point  i think he find it in general but i think we have to find it for each one  of equalibriom point  or we just find in general .
thus plot also are emazing did he use special code in matlab or mathematica? there is some app for plot thus kind of phase portrait but they not like this this is must have a special code for plotting in matlab or mathematica

> 

restart
> 

with(PDEtools)
> 

undeclare(prime, quiet)
> 

with(LinearAlgebra)
> 

declare(u(x, t), quiet); declare(U(xi), quiet); declare(V(xi), quiet)
> 

NULL
> 

DU := V

V

 (1)
> 

DV := U^2*beta/(delta*k^2)+(alpha*w/(delta*k^3)+1/(delta*k^2)-w^2/(delta*k^4))*U

U^2*beta/(delta*k^2)+(alpha*w/(delta*k^3)+1/(delta*k^2)-w^2/(delta*k^4))*U

 (2)
> 

var := U, V

U, V

 (3)
> 

J := Student:-MultivariateCalculus:-Jacobian([DU, DV], [var], 'output' = 'matrix')

Matrix(%id = 36893490902008029780)

 (4)
> 

Eigenvalues(J, implicit, output = 'list')

[RootOf(_Z^2*delta*k^4-2*U*beta*k^2-alpha*k*w-k^2+w^2, index = 1), RootOf(_Z^2*delta*k^4-2*U*beta*k^2-alpha*k*w-k^2+w^2, index = 2)]

 (5)
> 

Eigenvalues(J)

Vector[column](%id = 36893490902007993756)

 (6)
> 

NULL
> 

NULL
> 

DU1 := V

V

 (7)
> 

V = solve(DU1, V)

V = 0

 (8)
> 

DV1 := U^2*F[2]+U*F[1]

U^2*F[2]+U*F[1]

 (9)
> 

U = solve(DV1, U)

U = (0, -F[1]/F[2])

 (10)
> 

so*we*have*three*equalibriom*point*(0, 0)

so*we*have*three*equalibriom*point*(0, 0)

 (11)
> 

J1 := Student:-MultivariateCalculus:-Jacobian([DU1, DV1], [var], 'output' = 'matrix')

Matrix(%id = 36893490902007256836)

 (12)
> 

Eigenvalues(J1, implicit, output = 'list')

[RootOf(-2*U*F[2]+_Z^2-F[1], index = 1), RootOf(-2*U*F[2]+_Z^2-F[1], index = 2)]

 (13)
> 

Eigenvalues(J1)

Vector[column](%id = 36893490901938203092)

 (14)
 

NULL

Download bi-1.mw
Please Wait...