On this page, are a number of spreadsheets that can be used to help understand modeling.
1) Basic Dynamical System 1 equation for use with equations of form u(n)=a u(n-1) + b
2) Basic Dynamical System 2 equations for use with equations of form
u(n) = a1 u(n-1)
+ b1 v(n-1) + c1
v(n) = a2 u(n-1) + b2 v(n-1) + c2
Gives time graphs for u and v and uv-graph.
3) Weight simulation: You can experiment with different metabolic rates and different calorie consumptions. You can also add exercise as a component.
4) Using Goal
Seek in Excel. l
Click on "Tools", then "Goal Seek" on the Tools menu.
1) In first cell, give cell location that you want to equal some value,
2) In second cell, give value you want it to have.
3) In third cell, give cell location that you want to change to achieve this
goal. (Third cell cannot be a "locked" cell.)
5) Fitting Curve to Data: Given data, (0,u(0)), (1,u(1)), etc., this spreadsheet will help you fit lines to the data and find the equation for the line.
6) Second Order System: For use with dynamical systems u(n)=au(n-1)+bu(n-2)+c
7) Simulates drawing a bead from each of two bowls with events being independent.
8) Markov chains: Here we simulate drawing a bead from either the red or blue bowl, then draw the next bead from the bowl that is the same color as the last bead drawn. We keep repeating this process. The simulation helps us estimate the fraction of the draws that will be red and the fraction that will be blue?
9) Here are several roulette simulations. In the first roulette simulation, you can pick the probability of getting a red, normally 18/38, how much you start with, and your goal. The spreadsheet then "plays" roulette until you reach $0 or your goal. To simulate another game, just retype the starting amount. See how often you win. In the second roulette simulation, you can again pick your starting amount, goal, and probability of winning. A graph will display your results over the first 50 bets. In addition, it will simulate this game for a total of 100 times, giving you the number of times you reached your goal and went broke out of the 100 games. For this simulation, keep the goal low. The third roulette simulation is similar to the second, except the goal is 100 and the graph gives your standing over your first 3000 bets. (I would like to thank Deane Arganbright, who helped with the development of the second and third simulations.)
10) This spreadsheet will make time graphs and webs for nonlinear dynamical systems. Follow instructions for rescaling the graphs.
11) This is a series of genetic simulations In basic genetics spreadsheet simulation you will develop a basic understanding of genetic principles by seeing how the genetic makeup of a population changes over time, assuming no mutation or selective advantage. Eugenics spreadsheet simulates the change in genetic makeup over time given the application of negative eugenics. Malaria survivors spreadsheet simulates the number of survives given particular assumptions on the spread of sickle cell anemia and the spread of malaria. Sickle cell evolution simulates the evolution of the sickle cell allele under assumed risk of malaria. Mutation spreadsheet simulates the change in genetic makeup of a population assuming alleles mutate from one kind to another. Mutation/lethal trait spreadsheet simulates the change in genetic makeup given mutation toward a lethal trait.
12) This is a series of spreadsheets that make spirals of different kinds. The basic spiral is a sequence of lines in which each line has a length that is the same fraction of the length of the previous line, and turned counterclockwise by a given angle. For the next spreadsheet, an isosceles triangle is constructed with given base angles. A new isosceles triangle with same base angles is constructed with one of its legs being half the base of the previous triangle. Another spiral is constructed by embedding squares within squares, then coloring one portion of one side of each square. For the next spiral, a regular polygon is constructed with any given number of sides. Each side is bisected and bisection points are connected to form inscribed polygon. This is continued. Half of a side of each polygon is colored to form a spiral.
13) This spreadsheet constructs the Koch curve, as well as variations on the Koch curve in which an isosceles triangle is constructed on the middle of each line segment so that the lengths of the sides of the triangle equal the lengths of the line segments left on either side of the triangle. As the angle increases from 0 to 90 degrees, the dimension of the curve increases from 1 to 2. You can also use the Sierpinski spreadsheet to see the development of the Sierpinski triangle, in which middle quarters of triangles are removed. In this spreadsheet, the middle is not actually removed. You can find better versions of the Sierpinski triangle elsewhere on the internet.
13) Bifurcation diagram draws the bifurcation diagram for the logistic equation u(n+1)=u(n)+ru(n)[1-u(n)]. It can be adapted to draw bifurcation diagram for other dynamical systems.
There are numerous websites that have better constructions of the Koch curve, Sierpinski triangle, and bifurcation diagrams for the logistic equation u(n+1)=ru(n)[1-u(n)]