Transcript Bifurcation* - Mathematics - Missouri State University

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 longrun 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
Saddle-Node Bifurcation
dy
 a  y2
dt
Consider the critical points for
If a is positive…
y
-
a
 a
stable
+
-
unstable
y
If a is zero…
If a is negative…
0
-
there are no critical points!
semi-stable
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.
Pitchfork Bifurcation
dy
3
2
 ay  y  y (a  y )
dt
If a is negative or equal to 0…
If a is positive…
y
a
0
 a
+
+
y
stable
unstable
stable
stable
0
+
Pitchfork Bifurcation
Transcritical Bifurcation
dy
2
 ay  y  y (a  y )
dt
If a is negative…
y
If a is positive…
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