1. The problem statement, all variables and given/known data
Show that the following systems have a Hopf bifurcation at the origin as μ passes
through 0. What is the stability of the limit cycle created?
Determine the approximate amplitude and period of the limit cycle and interpret
this in terms of the original variables
2. Relevant equations
x˙ = 3μx−3y +x^2 + 3xy +yx^2
y˙ = x−μy −3y^3 −y^2x
3. The attempt at a solution
I assumed y*=sqrt(3)y, and get f = 3ux-sqrt(3)y*+(x^2)+3xy*+(x^2)y* and
g=-uy*+sqrt(3)x-xy*^2-sqrt(3)y*^3.
But I can't calculate the stability coefficient "a". I think I have made some errors in the f and g equation. Can anyone help me?
Show that the following systems have a Hopf bifurcation at the origin as μ passes
through 0. What is the stability of the limit cycle created?
Determine the approximate amplitude and period of the limit cycle and interpret
this in terms of the original variables
2. Relevant equations
x˙ = 3μx−3y +x^2 + 3xy +yx^2
y˙ = x−μy −3y^3 −y^2x
3. The attempt at a solution
I assumed y*=sqrt(3)y, and get f = 3ux-sqrt(3)y*+(x^2)+3xy*+(x^2)y* and
g=-uy*+sqrt(3)x-xy*^2-sqrt(3)y*^3.
But I can't calculate the stability coefficient "a". I think I have made some errors in the f and g equation. Can anyone help me?
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