Computational Simulations

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  1. Objective
  2. Motivation
  3. Priming questions
  4. Notes
    1. Introduction
    2. The Logistic Map
    3. Predictions
    4. Simulation
    5. Simulation cont.
  5. Questions
    1. Logistic Map in Octave

1. Objective

  • To show how simulation of Mathematical Models and ODEs are used in science

2. Motivation

  • It is a good time to start applying everything we have covered thus far

3. Priming questions

  • Watch parts 1 and 2 of the BBC's "The Secret Life of Chaos" [1]
  • In part 1 at 8:14, these equations were shown. What part of these equations look familiar?

4. Notes

4.1. Introduction

  • In this section, I will highlight several well-known Mathematical Models
  • There are many connections between these models and the material covered thus far

4.2. The Logistic Map

What is a "science model" that corresponds to this mathematical model (r is a number, e.g., 2.0)?

x(i+1) - x(i) = (r-1)*x(i) - r*x(i)*x(i)
  • The first term represents unconstrained population growth.
  • The second term represents a population decay.

4.3. Predictions

  • Predict what this model will predict

4.4. Simulation

  • Let's use a Spreadsheet to solve [2].
  • What should I do?

4.5. Simulation cont.

  • Solution: [3].

5. Questions

5.1. Logistic Map in Octave

  • Write an Octave program that reproduces the four figures shown on the Google Spreadsheet shown in class on this spreadsheet [4]. You may want to refer back to Introduction_To_Octave for information on creating plots.
  • Describe in 3-10 sentences what happens as r is increased above 3.5. The idea is to give a "science description" of the observation. That is, explain what you observe in the same way that you would when trying explain a painting to someone over the phone - the person on the other end should have a fairly good idea of what the main features of the painting are.

Extra Credit

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