1. Bifurcation * *Not to be confused with fornication “…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

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

5. 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. Saddle-Node Bifurcation

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

7. Pitchfork Bifurcation

8. Transcritical Bifurcation 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? y y If a is positive… If a is negative… stable unstable

9. Transcritical Bifurcation

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