Transcritical
Two branches persist while swapping their roles.
x′=rx−x²
Transcritical and Pitchfork Bifurcations
Exchange of stability and symmetry breaking
Different bifurcations reorganize equilibria in different ways: branches may exchange stability or split symmetrically.
watchone idea
→manipulateone example
→leave withone intuition
The essential idea: the shape and stability of equilibrium branches identify the mechanism of qualitative change.
One Concept · One Example
Transcritical and Pitchfork Bifurcations
Presented by Charlotte Moser · Coming soon
What to notice
- Transcritical branches cross and exchange stability.
- A supercritical pitchfork creates two stable states.
- A subcritical pitchfork exposes unstable side branches.
The mathematical structure
Pitchfork
Symmetry creates paired branches.
x′=rx∓x³
Compare three bifurcation geometries
Switch between the exact normal forms in the slides and move the parameter through the critical value.
Bifurcation diagrams distinguish stability exchange from symmetry-breaking branch creation.