Identify bifurcation and sketch diagram

Question

I am given the equation

$$\dot x = rx + \frac {x}{1+x^2} $$

I believe I've found fixed points at $x=0$ and $x= \sqrt \frac{-1-r}{r}$ and I think its stable for $-1<r<0$ but I'm not sure how to sketch this. I looked online but couldn't find any tools to do this. I also need to determine if it is a subcritical pitchfork or not. Thanks in advance.

Answer 1

You can easily determine the sign structure of the vector field on the one-dimensional phase space. The right side factors as $$ \dot x=x\left(r+\frac1{1+x^2}\right). $$

• For $r\ge0$ there are no roots from the second factor, and it is always positive. The resulting sign structure is $(-,+)$ for increasing $x$ values, $x=0$ is an unstable equilibrium.
• For $r<0$ the vector field points inwards if $|x|$ is large enough
  • for $-1<r<0$ the second factor has a pair of opposite roots, starting close to infinity for $r$ close to zero and meeting each other at $r=-1$ with a vertical tangent at $x=0$. The sign structure along the $x$ axis is $(+,-,+,-)$, meaning the outer equilibrium points are stable, the inner one at $x=0$ remains unstable.
  • for $r<-1$ the second factor is strictly negative, there is only one equilibrium at $x=0$. The sign structure is $(+,-)$, making this a stable equilibrium.