Download presentation

Presentation is loading. Please wait.

Published byKatelynn Christin Modified over 2 years ago

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

Similar presentations

OK

Mini-course bifurcation theory George van Voorn Part one: introduction, 1D systems.

Mini-course bifurcation theory George van Voorn Part one: introduction, 1D systems.

© 2018 SlidePlayer.com Inc.

All rights reserved.

Ads by Google

Pharynx anatomy and physiology ppt on cells Ppt on job rotation process Sources of energy for kids ppt on batteries Ppt on role of electronic media Ppt on geotextile Ppt on voluntary provident fund Ppt on solar energy projects in rural america Ppt on diodes and transistors Ppt on as 14 amalgamation and capital Ppt on diodes and transistors projects