Fwd: Great first week and unexpected progress on The Logistic Equation



Begin forwarded message:

From: Brenda McCaffrey <brendamc@asu.edu>
Subject: Great first week and unexpected progress on The Logistic Equation
Date: June 2, 2016 at 3:39:05 PM MST
To: Xin Wei Sha <xinwei.sha@asu.edu>, Pavan Turaga <pavan.turaga@asu.edu>, Christopher Roberts <christopher.m.roberts@asu.edu>, Brenda McCaffrey <brenda@bmccaffrey.com>
Cc: Todd Ingalls <Todd.Ingalls@asu.edu>

Hi all,

Thank you all again for this opportunity!  It was a fun week.

My plan for this week was to work through the computer project at the end of Ch. 1 of Chaos and Time-Series Analysis (Sprott, 2003) as an introduction to dynamical systems and a slide into Lyapunov equations.

The project for this week was to simulate the logistic equation and port it into Max, and possible play with it on the iStage.

I started with MathCad and ran through some tutorials and observed that the scripting language looked very object-oriented, so I scanned around for some examples of code, moved over to JavaScript, went through some tutorials there.  This led me to be curious about what Max/MSP could provide in terms of JavaScript training.

Well...some of you may know the answer.  The very first JavaScript tutorial in Max/MSP is for the logistic equation.  :-)

From reading ahead to Ch. 2 in Sprott, I was aware of the effect of changing the initial condition and also what I would call a fitting parameter (constant A in Sprott and r, the Malthusian parameter in other online documentation).  I looked at the behavior of the logistic equation and noted that it performs in the Max patch as expected from the discussion in Ch. 2 of the book.  I made a few modifications to the Max patch example so that I could interactively change the initial condition and the r value.

The output of the Max patch are tones.  As the simulation moves from one region into another (and eventually into Chaos for r = 4), the tones clearly provide an example of the range in question.

Ch. 2 of Sprott goes into much more detail about the logistic equation.  Here's my proposal for Week 2:

  • Review and summarize the behavior of the logistic equation from the reading.
  • The computer project at the end of Ch. 2 has to do with creating bifurcation diagrams of the logistic equation to illustrate that the period of the function doubles just prior to the onset of chaos -- I think it would be interesting to put this into an interactive iStage demonstration.  The thought would be to make it very simple.
    • Perhaps using XOSCs (left hand is initial condition, right hand is r?) so the user can experience and control the period doubling and onset of chaos by moving their hands.
    • The response would be sound.
  • The deliverable for the week would be the Max patch and a video demonstrating the operation.
Thoughts and suggestions?

Many thanks,
Brenda

Ch. 1 of "Chaos and Time-Series Analysis"

Hi Chris, Xin Wei and Pavan,


I'm home!  Sorry I missed your meeting yesterday.  We returned to Phoenix on Wednesday night and I was making some medical rounds yesterday.  We were on a very tiny cruise ship from Venice to Rome and I re-injured my foot in Croatia.  Rough seas for the ensuing days and I was unable to tender ashore.  No worries -- I'm super happy and had a great time!

The good news --- I had a blissful day anchored off Sicily with a view of Mt. Etna and read the first couple of chapters of Chaos and Time-Series Analysis.  Very rich and delicious book -- thanks, Pavan.  Lyapunov exponents are introduced in Ch. 5 and the chapters leading up to it are dense with thought-provoking implications of dynamical systems.  

My thought right now is to focus on a simulation for next week and create a little demo by next Thursday's Synthesis meeting.  At the end of each chapter, there is a simulation exercise and these build on each other leading to Lyapunov and beyond.  So, I could do the exercise in Chapter 1 for next week -- I think it would be a good way to get my feet wet and see where it leads.

For reference, here is the exercise:

1.8  Computer project:  The logistic equation
This project gets you started using the computer to model chaotic processes and to explore some of the more obvious properties of the logistic equation,....  The logistic equation isi perhaps the simplest example of a chaotic system.  It models a process that exhibits initial exponential growth with a nonlinearity that ultimately stops the growth.  Most of the common features of chaos are manifest in this simple example.  The variable X is advanced successively in discrete time steps denoted by Xn for n = 0,1,2,....  Start with an initial condition of X0 = 0.1 and iterate the logistic equation many times.  {This goes on for several paragraphs}

There are several steps to the exercise and I'll go through as many as possible and devise a small scenario to play out on the iStage using Max, and report on the experience and any results.

I'm sure this kind of thing has been done before, but I think it will be a good introduction and will require that I can play with the simulation, create a Max sub-patch, and create something experiential with it.  I hope it will be interesting.

Please let me know what you think.  I'll also post this and future notes on the dynamical-systems Posthaven site and on the Slack channel.

Good to be home!  Please feel free to share this with anyone who may be interested.

Regards,
Brenda

p.s., I think Monday is a holiday, so I'll be in Matthews on Tuesday morning at 10am.