Computational Simulations

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Contents

  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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