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Locate any bifurcation in 2D system

  • Thread starter fwang6
  • Start date
  • #1
6
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Homework Statement



bifurcation for the following 2D system:

Homework Equations



x'=ux−y+x^3,y′=bx−y

The Attempt at a Solution


I have got ux−y+x^3=0, y=bx, then x=0 and ±sqrt(b−u).

But I don't how to continue to find the bifurcation?
 

Answers and Replies

  • #2
pasmith
Homework Helper
1,735
410

Homework Statement



bifurcation for the following 2D system:

Homework Equations



x'=ux−y+x^3,y′=bx−y

The Attempt at a Solution


I have got ux−y+x^3=0, y=bx, then x=0 and ±sqrt(b−u).

But I don't how to continue to find the bifurcation?
What happens when [itex]b - u < 0[/itex]?

In general, you are looking for values of the parameters for which at least one eigenvalue of [tex]
\begin{pmatrix}
\frac{\partial x'}{\partial x} & \frac{\partial x'}{\partial y} \\
\frac{\partial y'}{\partial x} & \frac{\partial y'}{\partial y}
\end{pmatrix}
[/tex] at a fixed point has zero real part.
 
  • #3
6
0
if b-u<0,no critical points exist.
 

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