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Question: Eigenvalues Of Critical Points

Question:Eigenvalues Of Critical Points

trace 75 Maple
eigenvalues
May 22 2013
1

Hello guys ,

I found critical points for a 3-D dynamical system (x,y,z) and m which in general is not constant (m(z/y) is a function of z.y, when i calculate eigenvalue for every critical point ,the maple assume m constant . how i find eigenvalues for general m which is not constant and is a function of z/y .

Thank You

ww.mw

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restart
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with(DEtools):
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with(plots):
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f := proc (x, y, z) options operator, arrow; -1-z-3*y+x^2-x*z end proc;

proc (x, y, z) options operator, arrow; -1-z-3*y+x^2-x*z end proc

  

proc (x, y, z) options operator, arrow; x*z/m-2*z*y+4*y+x*y end proc

  

proc (x, y, z) options operator, arrow; -x*z/m-2*z^2+4*z end proc

 (1)
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sys := diff(x(t), t) = f(x(t), y(t), z(t)), diff(y(t), t) = g(x(t), y(t), z(t)), diff(z(t), t) = h(x(t), y(t), z(t));

diff(x(t), t) = -1-z(t)-3*y(t)+x(t)^2-x(t)*z(t), diff(y(t), t) = x(t)*z(t)/m-2*z(t)*y(t)+4*y(t)+x(t)*y(t), diff(z(t), t) = -x(t)*z(t)/m-2*z(t)^2+4*z(t)

 (2)
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solve({f(x, y, z) = 0, g(x, y, z) = 0, h(x, y, z) = 0}, {x, y, z});

{x = -(2*m-2)/(1+2*m), y = -(4*m-1)/(m+2*m^2), z = -(1-4*m^2-3*m)/(m+2*m^2)}, {x = 3*m/(m+1), y = -(1+4*m)/(4*m+2*m^2+2), z = -(-4*m-1)/(2+2*m)}, {x = 0, y = -1, z = 2}, {x = -4, y = 5, z = 0}, {x = -1, y = 0, z = 0}, {x = 1, y = 0, z = 0}

 (3)
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J := unapply(VectorCalculus:-Jacobian([f(x, y, z), g(x, y, z), h(x, y, z)], [x, y, z]), x, y, z):
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LinearAlgebra:-Eigenvalues(J(1, 0, 0))

Vector[column]([[2], [5], [(4*m-1)/m]])

 (4)
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LinearAlgebra:-Eigenvalues(J(-1, 0, 0))

Vector[column]([[-2], [3], [(1+4*m)/m]])

 (5)
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LinearAlgebra:-Eigenvalues(J(0, -1, 2))

Vector[column]([[-3], [(1/2)*(-3*m+(25*m^2-16*m)^(1/2))/m], [-(1/2)*(3*m+(25*m^2-16*m)^(1/2))/m]])

 (6)
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LinearAlgebra:-Eigenvalues(J(-(2*m-2)/(1+2*m), -(4*m-1)/(m+2*m^2), -(1-4*m^2-3*m)/(m+2*m^2)))

Vector[column]([[-(8*m^2-2+3*m)/(m*(1+2*m))], [-(4*m-1)/m], [-2*(-1+m^2)/(m*(1+2*m))]])

 (7)
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LinearAlgebra:-Eigenvalues(J(-4, 5, 0))

Vector[column]([[-5], [-3], [4*(m+1)/m]])

 (8)
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LinearAlgebra:-Eigenvalues(J(3*m/(m+1), -(1+4*m)/(4*m+2*m^2+2), -(-4*m-1)/(2+2*m)))

Vector[column]([[3], [(1/4)*(-3*m+(-31*m^2+256*m^4-16*m+160*m^3)^(1/2))/(m*(m+1))], [-(1/4)*(3*m+(-31*m^2+256*m^4-16*m+160*m^3)^(1/2))/(m*(m+1))]])

 (9)
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``



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