Animating Bifurcations
Animating Bifurcations
Eric Marland
Appalachian State University
27 August 2010
Animating Bifurcations
Welcome to Project Mosaic
We are a community of educators working to develop new ways to
introduce mathematics, statistics, computation and modeling to
students in colleges and universities.
Animating Bifurcations
The Purpose of Project Mosaic
To share ideas and resources to improve teaching and assessment
in mathematics, statistics, computation, and modeling.
And to make it really easy to do it.
Animating Bifurcations
Project Mosaic Mailing lists
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Moderated list for announcements and information of interest
to the Project Mosaic community. (Upcoming M-casts are
announced here, for example.)
discussion@mosaic-web.org
Unmoderated list where list members can discuss ideas and
share resources to improve teaching and assessment in
mathematics, statistics, computation, and modeling.
Visit http://mosaic-web.org to join one or both lists.
Animating Bifurcations
Upcoming M-casts
September 10: Vittorio Addona, Helping Students Understand
Regression Coefficients: An Example of Modeling Body Fat
Percentage
Time: 11:30 CDT (12:30 Eastern, 9:30 Pacific)
September 24: Nicholas Horton, Being Warren Buffett: a
classroom and computer simulation of financial risk
Time: 11:30 CDT (12:30 Eastern, 9:30 Pacific)
Animating Bifurcations
Learning Goals
1
Understand the difference between parameters and variables
2
Develop comfort level with exploring the software (Maple)
3
Differentiate between qualitative and quantitative changes
4
Strengthen understanding of phase planes and the trace-det
plane
Animating Bifurcations
Fitting into Mosaic
Visualization and Parameter sensitivity analysis
Acquiring an exploratory comfort level with software
Gain student interest
Translates to other courses
Animating Bifurcations
Prelude to a Bifurcation - The Lorenz Equations
1
Animate the solution as a function of time in such a way that
we can watch the solution trace out its path.
2
Animate the rotation of the axis for a complete solution so
that the three dimensional structure is revealed.
3
Possibly do both animations at the same time. You may find
that doing both at the same time is distracting and that it is
easier to see what is going on by doing them separately.
Animating Bifurcations
The Animation Assignment
1
Create a phase plane plot of several exemplary solutions and
the direction field.
2
Animate the changes in those solutions (same initial
conditions) as a parameter is changed.
Each group is assigned a system of differential equations or a task
to create a system that has a particular type of bifurcation.
Animating Bifurcations
Concrete Examples
dY
=
dt
dY
=
dt
a −1
2
0
Y
a a2 + a
1
a
dY
=
dt
dY
=
dt
(from Blanchard, Devaney, and Hall)
a 1
a a
a a
1 0
Y
Y
Y
Animating Bifurcations
Exploratory Examples
Find a system that moves from a stable spiral to a stable node
and back to the same location in the trace determinant plane,
but with different values.
Find a system that begins with a large positive trace and ends
with a large negative trace.
Find a system that begins with a positive determinant and
ends with a negative determinant.
Find a system that move diagonally though the origin in the
trace determinant plane.
Find a system that moves ”around the world”, moving
completely around the origin and back to the original location
in the trace determinant plane.
Animating Bifurcations
The basic approach
Thrown in the pool, variations on a theme
Animating Bifurcations
Grading Issues
Animating Bifurcations
Expansion and Extension
Animations for calculus
Other software packages
...
Animating Bifurcations
Sample Code
>with(DEtools):with(plots):
>eq1:=diff(x(t),t) = 4*x(t) - 2*y(t):
>eq2:=diff(y(t),t) = -3*x(t) + i/10*y(t):
>p:=seq(DEplot({eq1,eq2},{x(t),y(t)}, t=0..5,
x=-10..10, y=-10..10, [[x(0)=4,y(0)=8.7445]],
linecolor=blue,numpoints=300),i=-20..30):
>display(p,insequence=true);
Animating Bifurcations
And the Result (one of the frames).
Animating Bifurcations
What materials are going to be available on the website?
These slides
Actual assignments
Sample code, several variations
Added code from you ...
Sample videos from students
Animating Bifurcations
For additional information or to sign up to give an MCAST visit,
http://mosaic-web.org/