x=rx-x1+x^2 bifurcation

The equation x'=rx-x/1+x^2 has a bifurcation at x=0.

To see this, we can look at the sign of x' on either side of x=0.

For x>0, x' is positive if r>0 and negative if r<0. This means that if r>0, solutions will move away from x=0, while if r<0, solutions will move towards x=0.

For x<0, x' is negative if r>0 and positive if r<0. This means that if r>0, solutions will move towards x=0, while if r<0, solutions will move away from x=0.

Therefore, the behavior of solutions near x=0 depends on the sign of r. This is a bifurcation, as a small change in the parameter r can lead to a qualitative change in the behavior of solutions