Drawing Bifurcation diagrams for a dynamical system

[Phyrrus]

1. Homework Statement [/b]
Consider the dynamical system
\frac{dx}{dt}=rx-\frac{x}{1+x}
where r>0
Draw the bifurcation diagram for this system.

Homework Equations

The Attempt at a Solution

Well fixed points occur at x=0,\frac{1}{r}-1 and x=0 is stable for 0<r<1 and unstable for all r>1
For the fixed point at \frac{1}{r}-1, however, it is stable when r-r^{2}<0

So how do I draw the non-zero bifurcation points? Thanks

 

[jackmell]

1. Homework Statement [/b]
Consider the dynamical system
\frac{dx}{dt}=rx-\frac{x}{1+x}
where r>0
Draw the bifurcation diagram for this system.

Homework Equations

The Attempt at a Solution

Well fixed points occur at x=0,\frac{1}{r}-1 and x=0 is stable for 0<r<1 and unstable for all r>1
For the fixed point at \frac{1}{r}-1, however, it is stable when r-r^{2}<0

So how do I draw the non-zero bifurcation points? Thanks

Well, the bifurcation diagram is a diagram illustrating how the fixed points change as a function of the parameter. So you have two fixed points:

(0,\frac{1-r}{r})

So those change as a function of r right? Got two until r=1, got one there, then two again when r>1. Can you not just plot those two fixed points as a function of r? You can do that.