# Locate any bifurcation in 2D system

## 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?

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pasmith
Homework Helper

## 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.

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