Show that system is Transcritical bifurcation

Question

In what ways can you show that transcritical bifurcation occurs? For example take the system

$$\dfrac{dx}{dt}=xr+2x^2 $$

Answer 1

Hints:

We allow $r$ to vary and it can be less, equal or greater than zero.

For these three ranges, we have the critical points:

• $r \lt 0 \rightarrow -r x + 2x^2 = 0 \implies x = 0, x = \dfrac{r}{2}$
• $r = 0 \rightarrow 2x^2 = 0 \implies x = 0$
• $r \gt 0 \rightarrow r x + 2 x^2 = 0 \implies x = 0, x = -\dfrac{r}{2}$

Note that for all three choices of $r$, $x = 0$ is a critical point.

Now, we can plot a phase portrait ($x' = rx + 2x^2, y' = -y$) for $r \lt 0, r = 0, r \gt 0$ to determine the stability of the critical points.

You should be able to determine the stability of each critical point and then draw the bifurcation diagram in the $rx-plane$. Here is a start. Add in the dashed lines and the stability information.