Bifurcation *

1. Bifurcation* “…a bifurcation occurs when a small smooth change made to the parameter values of a system will cause a sudden qualitative change in the system's long-run stable dynamical behavior.“ ~Wikipedia, Bifurcation theory *Not to be confused with fornication

3. For an equation of the form Where a is a real parameter, the critical points (equilibrium solutions) usually depend on the value of a. As a steadily increases or decreases, it often happens that at a certain value of a, called a bifurcation point, critical points come together, or separate, and equilibrium solutions may either be lost or gained. ~Elementary Differential Equations, p92

4. y y Saddle-Node Bifurcation Consider the critical points for - If a is positive… stable + unstable - - If a is zero… semi-stable - If a is negative… there are no critical points!

5. Saddle-Node Bifurcation If we plot the critical points as a function in the ay plane we get what is called a bifurcation diagram. This is called a saddle-node bifurcation.

6. y - y + - - + + Pitchfork Bifurcation If a is negative or equal to 0… If a is positive… stable unstable stable stable

7. Pitchfork Bifurcation

8. y y Transcritical Bifurcation If a is negative… If a is positive… stable stable unstable unstable Note that for a<0, y=0 is stable and y=a is unstable. Whenever a becomes positive, there is an exchange of stability and y=0 becomes unstable, while y=a becomes stable. Cool, huh?

9. Transcritical Bifurcation

10. Laminar Flow Low velocity, stable flow High velocity, chaotic flow