Pitchfork Bifurcation

Supercritical Pitchfork Bifurcation

Normal form: $\dot{x}=rx-x^3$

• pitchfork bifurcation is expected in systems that show invariance ($\varphi \to -\varphi$)

Subcritical Pitchfork Bifurcation

Has a normal form $\dot{x}=rx+x^3$. Subcritical pitchfork bifurcations look like inverted supercritical ones, where initially a stable fixed point at $x^{\text{*}}=0$ exists with two unstable branches when $r<0$. $r=0$ is a bifurcation point where the origin becomes the only fixed point and unstable additionally. The $x^3$ term helps with instability and makes blow-up possible, where $x(t)\to \pm \infty$ in finite time.

However, in application these systems often introduce a stabilising term (such as + $x^5$) that generates stable branches above and below the unstable branches of the subcritical pitchfork bifurcation such that hysteresis occurs, where increasing $r$ from $a$ to $b$ and then back to $a$ results in $x(t)$ at a different position then at $a$ before being moved.