How can I determine the type of bifurcation I have found?

Question

I was tasked with finding and classifying the bifurcation of the following system:

$$\frac{dx}{dt} = rx(x+1)-1$$

I have found that the bifurcation occurs when $r=1$, with equilibrium points at $x=0$ and $x=\frac{1}{r}-1$, but I am not sure how to determine the type of bifurcation it is. I have created a plot of the equilibrium points against r, but it looks nothing like any other plots I have seen that give the type of bifurcation. Is there a way to find the type of bifurcation it is analytically, or how can I otherwise determine the type of the bifurcation?

Answer 1

The equilibrium points satisfy $x^2 + x - 1/r = 0$, with solutions $$ x_\pm = -\frac{1}{2}\pm\sqrt{\frac{1}{4}+\frac{1}{r}}\, . $$ The Jacobian $J(x) = r (2x + 1)$ evaluated at the equilibrium points reads \begin{aligned} J(x_\pm) &= \pm r\sqrt{{1}+{4}/{r}} \\ &= \pm \text{sgn}(r)\sqrt{r(r+{4})}\, , \end{aligned} where $\text {sgn}$ denotes the sign function. Therefore,

• if $r>0$, $x_+$ is an unstable equilibrium while $x_-$ is asymptotically stable;
• if $-4<r<0$, there is no equilibrium;
• if $r<-4$, $x_-$ is an unstable equilibrium while $x_+$ is asymptotically stable.

Two bifurcations occur: one at $r=0$ and one at $r= -4$, as illustrated on this diagram:

One can remark that these bifurcations are of saddle-node type.