Math 2030

Math 2030 Fall 2012

Instructor: Office: Hours: Office Phone: e-mail:

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Peter R. Wolenski 326 Lockett Hall M 10:30-11:30, W 11:30-12:30, F 10:30-11:30. 578-1606 [email protected]

Introduction

1.1

The subject matter

The title of this course is Discrete Dynamical Systems, but what does that mean? •

Discrete: Refers to counting time steps in contrast with continuous time. Continuous time systems involve differential equations and is considerably more complicated.

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Dynamical: This means there is a time component.

Something is dynamical,

i.e. changing or evolving over time. •

Systems: A single system contains the objects and rules that govern the dynamic movement. However, typically we are not interested in just one system, but rather a family of parameterized systems. The main topic of this course is to study the qualitative properties of this family as the parameter changes.

A Discrete Dynamical System consists of a set X (called the state space) and a function F (·) (called the dynamic map) that maps X to X. This is notated by F (·) : X → X. Given x0 ∈ X (called the seed), one defines the iterates of x0 by  x1 = F (x0)     x2 = F (x1) = F (F (x0))     x3 = F (x2) = F (F (F (x0)))   .. .. .. . . . hx0i =   x = F (x  k+1 k ) = F (. . . (F (x0 )) . . . )  | {z }    n+1 times    .. .. .. . . . The entire sequence hx0 i = {x0 , x1, x2, . . . } is called the orbit or trajectory originating from x0, and one should think of the iterates as the successive states to which the trajectory moves. The main topic of this course is to study and understand the asymptotic behavior 1

of trajectories; that is, what happens to xk as k → ∞. This seems (and is!) simple, but achieving a deeper understanding requires the mastery of the two facets of this course: (I) Theory - mathematically rigorous proofs; and (II) Experiments - graphs and numerics generated by computer software. But in fact it is not one system we are generally interested in, but rather an entire family of systems that typically depend on a parameter λ. Thus we have a map F (·, ·) : X × Λ → X in which for each λ ∈ Λ gives rise to a system using F (·, λ) := Fλ(·) : X → X. The interesting topic of bifurcation analysis is to see how the behavior of orbits change as λ changes. The course is also about Chaos and Fractals. The three topics - discrete dynamical systems, chaos, and fractals - are three interconnected aspects of the same subject. So where to begin? We choose by starting with discrete dynamical systems and will see how the other concepts arise naturally in this study. This course is a good opportunity to become familiar with mathematical software that facilitates computation and visualization, and we shall mainly use Excel and Mathematica. These will suffice for the course, although you are welcome to use Maple and/or Matlab and/or Mathcad and/or anything else. I emphasize that this is a mathematics and not a computer science course, but a computer is fundamental to the material even though no a priori computer skills are required. The only prerequisite is Calculus I (Math 1550). However, it is mathematics of a different sort, and in fact, this course is meant to be a so-called bridge course, which means that the material involves abstract concepts and precise logical reasoning (in contrast to most of the calculation-based material in your previous courses). A computer will assist and reinforce comprehension of the abstractions, but in fact, the entire subject was only developed to the extent we see it today because of the ability and accessibility to easily compute complicated objects. You should learn the basics of mathematical software, a skill which will help you throughout your career since it has become crucially important in every field of scientific and technological study.

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Notes and software Your notebook

The course requires no text book, and instead will follow my lecture notes. These are (and future notes, exams, etc. will be) posted at http://www.math.lsu.edu/∼wolenski/Math2030 The notes are largely based upon Devaney’s text “A First Course in Chaotic Dynamical Systems” which we have used in the past. You should carefully read these notes, and understand every detail and do every exercise at the end and those sprinkled among the text. But you must also keep your own notebook which should be at your side while reading, and make additional notes to yourself. Any question or confusion should be brought up in class. 2

1.2.2

Software

As already mentioned, we shall use some standard software. There are several reasons for doing so, and perhaps the most important is the benefit you gain in learning it. It’s OK if you know next to nothing about computer programming, for you should be able to easily pick up what you need to know for this course from the lecture notes. If you know a lot about computing, then you will enjoy devising ways to put that knowledge to use. You probably know some Excel, for if nothing else, you should use it to keep track of your bank and checking balances. If not, oh well, we will teach you the basics. I have previously and primarily used the software Maple, but this is not currently available to students as it used to be. (It is in the virtual laboratory and you can use it there.) Thus we will mainly go with Mathematica, a software available to all of you from Tigerware, and which has an amazing functionality (if at times an ugly and confusing syntax, but you get used to it). A site that has relevant downloadable Mathematica notebooks can be found at http://library.wolfram.com/infocenter/MathSource/549/

1.3

Grading

Your final grade will be based on the following: Notebook (≈500 points): This is the most important part of the class, and you will not pass if you do not comply with the mandate that you are required to keep a notebook. This will serve as your journal and will record your participation in the class. The notebook should contain your class notes, homework assignments, exercises, and miscellaneous comments. I will collect them roughly every three weeks (five times for the semester). Quizzes (≈50 points): Short in-class reinforcements of definitions, calculations, and some proofs. You will be asked to volunteer to present material to the Expositions (≈50 points): class. Midterm exam (100 points): An in class exam to mark your progress. Course project (100 points): A research-based report, computer-based simulation or project, or service learning lecture to a high school math class of your choice. More details to be given later. Final exam (200 points): Comprehensive Overview.

1.4

Web sites

There are lots of interesting web sites out there, from tutorial lecture notes to programs that construct fractals to advanced chaotic dynamical systems theory. The following list is somewhat dated, and I am sure there are many much better sites these days. Please pass along sites that you find helpful. I. Discrete Dynamical Systems 3

A. http://math.bu.edu/people/bob/ This is Bob Devaney’s website. B. http://archives.math.utk.edu/topics/dynamicalSystems.html General links to a wide assortment of topics in dynamical systems. C. http://www.cs.brown.edu/research/ai/dynamics/tutorial/home.html A general tutorial surveying many modern applications. D. http://www.ba.infn.it/ zito/plaw.html Essays on aspects of Dynamical Systems. II. Software A. Download Mathematica from tigerware: http://tigerware.lsu.edu/list.aspx?id=172. Unfortunately Maple can apparently only be used by students in the virtual lab. B. http://www.wolfram.com/products/mathematica/index.html The home page for Mathematica; tutorials are located at http://library.wolfram.com/tutorials http://www.math.utah.edu/lab/ms/maple/maple.html C. http://www.indiana.edu/ statmath/math/mma/gettingstarted/index.html A more user-friendly tutorial from the University of Indiana to get started. III. Fractals A. http://www.ultrafractal.com/ A site to create your own fractals. B. http://www.parkenet.org/jp/ufresources.html All sorts of resources devoted to fractals.

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Overview

The following is a broad outline of the course material. Part I. Discrete Dynamical Systems 1. Introduction and math background. 2. Definitions, examples, and numerical iteration. 3. Attractors and repellers. 4. The quadratic family and bifurcations Part II. Chaos 1. Cantor’s Middle Third Set 2. Symbolic dynamics and abstract spaces 3. Chaos and Sarkovskii’s Theorem Part III. Fractals 1. Self-similar sets and fractal dimension 2. Complex dynamics 3. The Julia and Mandelbrot sets

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