saddle-node bifurcation

• fp appears “out of blue” - no fixed points (ghost/bottleneck) -> 1 point -> 2 points (saddle and stable)

Example I: Genetic control

• Stable manifold that separates them into 2

Example III:

• Look at symmetries! ()

Hopf bifurcation

• can only happen in 2D or higher

Supercritical Hopf bifurcation

• For exam

Subcritical hopf bifurcation

• error on slide with $\dot{r} = 0 \dots$
• important is that as $\mu$ gets lower (under 0) it splits into stable center and unstable circle (the other circle doesn’t matter)

Poincare maps

example

• start from right side $$ \int_0^{2\pi} d \theta = 2 \pi $$