“…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
For an equation of the form
f ( a, y )
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
Consider the critical points for
If a is positive… -
If a is zero…
If a is negative… there are no critical points!
If we plot the critical points as a function in the
ay plane we get what is called a bifurcation
This is called a saddle-node bifurcation.
ay y y(a y )
If a is positive… If a is negative or equal to 0…
0 unstable 0 stable
a - stable
ay y y (a y )
If a is positive… If a is negative…
0 unstable a 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?
Low velocity, stable flow
High velocity, chaotic flow