# Bifurcation *

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```					                                       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
For an equation of the form

dy
 f ( a, y )
dt

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
dy
Consider the critical points for
 a  y2
dt
y
If a is positive…                               -
a             stable

+
 a                unstable
-
y

If a is zero…
-
0                          semi-stable
-

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
diagram.

This is called a saddle-node bifurcation.
Pitchfork Bifurcation

dy
 ay  y  y(a  y )
3         2

dt
If a is positive…                If a is negative or equal to 0…
y                                 y
-              stable
a
+                                   -
0                   unstable       0                             stable

     a    -            stable
+
+
Pitchfork Bifurcation
Transcritical Bifurcation
dy
 ay  y  y (a  y )
2

dt
If a is positive…                        If a is negative…
y
y

a                 stable
0                stable

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?
Transcritical Bifurcation
Laminar Flow

Low velocity, stable flow

High velocity, chaotic flow

```
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