- What is a difference equation?
- Stair-step diagrams.
- Geometry of linear maps
- Stability tpyes of fixed points.
- Linearization theorem.
- Application: Why Newton's method converges
fast for good starting values?
- When the Linearization does not suffice:
Using PHASER, determine if the origin is an unstable, stable,
or asymptotically stable fixed point of the following maps.
Make sure to include one screen image of stair-step diagram
for each map. Show that the origin is a fixed point and the derivative of each equation
at the origin is either 1 or -1. Now explain why the Linearization theorem
has nothing to offer when the derivative at a fixed point is 1 or -1.
Hint: Enter the MAP x1 -> a*x1 + b*x1*x1*x1 into PHASER. Now set
the parameter values appropriately to study any one of the maps above.
- x1 -> x1 - 1.15*x1*x1*x1
- x1 -> x1 + 1.15*x1*x1*x1
- x1 -> -x1 - 1.15*x1*x1*x1
- x1 -> -x1 + 1.15*x1*x1*x1
- As a parameter is varied:
Consider the map x1 -> a + x1 + x1*x1 where a is a parameter.
Find three values of the parameter a for which the map has 0, 1, or 2
fixed points. Determine the stability types of these fixed points.
You should enter this map into Phaser as a Custom equation. Then using the
stair-step diagrams try to find the requested parameter values.
How much can you borrow?:
Suppose you borrow x0 dollars from a bank at 2 per
cent per month interest,
and that you pay your bank 40 dollars per month.
Derive the difference equation that models your debt to the bank at the end of each month.
(Hint: the equation you need is a linear equation. So
use the Linear 1D in the Map library of PHASER
and set the parameters appropriately.)
If your original debt is 580 dollars, how long will it take you to pay the loan?
If your original debt is 3600 dollars, how long will it take you to pay the loan?
What is the maximum amount you can barrow?
Beverton-Holt Stock-recruitment model:
xn+1 = rxn/[1 + ((r - 1)/k)xn]
This is a biologically important fisheries model containing
two parameters r (growth rate) and k (carrying capacity).
Despite its complicated form, this model has simple dynamics.
We assume that both parameters take on non-negative values.
Enter this model into Phaser.
To understand the geometric meanings of the parameters,
fix k (say at 1.2) and vary r make a Gallery of growth curves using
the stair-step view of PHASER (study the Phaser Tutorials Lesson 6
and Lesson 13 to learn about the Gallery and making SlideShow).
You may want to take big window size
(-1 , 15; -1, 15) to see what happens as the population gets large.
Next, fix r = 1.45 and vary k. Describe biologically what you observe in these
Find the fixed points of the model as a function of the parameters.
For what ranges of the parameters they are biologically significant?
Determine the stability type of the fixed points, using the
Linearization Theorem. Can the positive fixed point
become unstable as the parameter r or k is increased?
- Write a summary of the possible dynamics of a population descibed by
the Beverton-Holt model and interpret your findings in biological terms.
- Is faster better?:
Consider the map x1 -> 0.375*x1 + 1.5/x1 - 0.5/(x1*x1*x1).
Iterate this map (derived by Alkalsadi around 1450AD)
with the starting value 1.6 until 15 digits settle down.
How many iterations are required? Now, count the number of all the
additions, subtractions, multiplications and divisions you have made to obtain
your approximation to sqrt(2).
Next do the same with Newton's method using x1 ->0.5*(x1 + 2.0/x1). Which method requires
fewer number of iterations? Which method requires fewer
number of operations to compute sqrt(2) to the same precision?