A Hands-On Laboratory For Chaotic
Systems
Frank
SEVERANCE and DAMON MILLER,
June 9-11, 2008 in
Note: There is a course fee of $250 (in addition to the application fee) and
a materials fee of $100 for components, which participants will be able to take
with them in the form of completed project kits.
Classical
modeling and analysis, especially at the undergraduate level, is almost always
done assuming a linear model. This bias
usually trains students to believe that nonlinear systems are rare, are to be
avoided and are difficult to work with.
This workshop will address these misconceptions by exploring deceptively
simple, yet intriguing chaotic electronic circuits. Chaotic systems produce bounded, non-periodic
oscillations. Trajectories are confined
to bounded spatial regions where all fixed-points are unstable, thus creating a
set of “strange attractors,” where trajectories tend to dance in a seemingly
random fashion.
Chaotic system demonstrate extreme sensitivity to initial
conditions, thereby demonstrating the so-called “butterfly effect”. For these reasons, such systems demonstrate
seemingly random behavior, but are obviously deterministic in that they are
(most often) defined by differential or difference equations, whose solution is
unique. These properties are certainly
outside the experience of most undergraduate courses, where unique and well
behaved steady-state behavior of all systems is understood to be the case, if
not explicitly assumed.
The
goal of this workshop is to study chaotic systems on multiple levels; however,
participants are not assumed to have a formal background in nonlinear
systems. Thus, we will first explore
some of the characteristic behaviors of first, second and third order
continuous and discrete nonlinear systems.
Since there is in general no means by which to explicitly solve such
systems, we will use the qualitative approach developed by Henry Poincaré to
characterize the nature of their solution.
Next we will use simulation methods to observe the trajectories of these
systems. Then we will design and
construct physical realizations of the Lorenz and logistic map chaotic systems
as electronic circuits. Finally, using data sampling techniques we will analyze
the circuits to characterize their chaotic behavior. The workshop will close with a discussion of applications
and pedagogical approaches to nonlinear phenomena in general and chaos in
particular for the undergraduate science classroom.
Participants
in the workshop will each receive electronic kits to construct the Lorenz and
logistic map systems as electronic circuits.
During the workshop they will build and utilize these kits and receive
instruction in applying data acquisition techniques with a data acquisition
board and software. This experience will
enable participants to demonstrate chaotic systems at their home institutions.
For college teachers of: all quantitative sciences. Mathematicians, computer scientists and
engineering, physics and chemistry faculty will find it especially appropriate
for their classrooms. Prerequisites: A background in basic
differential equations, while desirable, is not required. No background in electronic circuits is
necessary. Limit: 16 participants.
Professors Severance and
Miller both are faculty members of the Department
of Electrical and Computer Engineering at