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 $$\begin{pmatrix} \frac{\partial x'}{\partial x} & \frac{\partial x'}{\partial y} \\ \frac{\partial y'}{\partial x} & \frac{\partial y'}{\partial y} \end{pmatrix}$$ at a fixed point has zero real part.

 

[fwang6]

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