- Announcements:
- Homework for section 6.5: p. 498, #22-25; due 11/14.
- Last time Dr. Jim Cushing, Professor of Mathematics at the
University of Arizona, provided us with the elemental aspects of
Dynamics, our current unit.
- Today:
- Dr. Cushing introduced a fair amount of vocabulary: describe the following:
- State variable
- Initial value
- Iterative rule
- Sequence of numbers
- Deterministic behavior
- Parameter
- Bifurcation
- Linear rule
- Non-linear rule
- quadratic polynomial
- What was a surprising consequence of the interest bearing problem?
(We'll talk more about financial math in our personal finance unit,
which will accompany similar analyses.)
- Describe what we saw in the flour beetle data. What was his point
(the "punch line") about the 95% culling rate?
- What were some of the conclusions that Professor Cushing drew?
- Non-linear versus linear thinking
- Surprising consequences of simple rules
- "Tipping points" are more common than we think.
- Chaos is roughly defined as "sensitive dependence on
initial conditions", but what this means is that if you start at two different
points in a system (for example, in weather you might be thinking of a
state of three variables, temperature, barometric pressure, and wind
speed), very close to each other but distinct, then eventually the system will
produce wildly different behaviors (even those the starting conditions were
extremely close).
- Dr. Cushing implied that this leads to a serious problem: although
the systems are deterministic, if we screw up at all and give
the wrong initial conditions (e.g. through measurement error), then
after a short while we may have a prediction which has nothing to do
with the actual value.
- Let's try an exercise involving calculators and the quadratic function
which illustrates bifurcation
- Try values of b=1.75, 3.3, 3.5, 3.9
- Try to find an 8-cycle
- Let's try the exercise involving calculators and the SIN function
which illustrates "sensitive dependence on initial conditions".
- Lorenz's Butterfly.
- Some links:
- Cobwebbing - an example (#6)
Website maintained by Andy Long.
Comments appreciated.