Locate any bifurcation in 2D system

[fwang6]

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?

 

[pasmith]

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 b - u < 0?

In general, you are looking for values of the parameters for which at least one eigenvalue of <br /> \begin{pmatrix}<br /> \frac{\partial x&#039;}{\partial x} &amp; \frac{\partial x&#039;}{\partial y} \\<br /> \frac{\partial y&#039;}{\partial x} &amp; \frac{\partial y&#039;}{\partial y}<br /> \end{pmatrix}<br /> at a fixed point has zero real part.

 

[fwang6]

if b-u<0,no critical points exist.