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The Department of Mathematics and Statistics proposes the following course for inclusion in University Studies, Unity and Diversity, Critical Analysis at Winona State University. This was approved by the full department on April 5, 2001.

Course: Chaos Theory (MATH 315)  3 s.h.

Catalog Description: An introduction to chaos theory and fractal geometry. Topics will include bifurcations, Julia sets, the Mandelbrot set, fractal geometry, iterated function systems and a survey of the applications of this theory to a variety of disciplines. Prerequisite: MATH 160.

Name: Barry A. Peratt

Title: Assistant Professor of Mathematics and Statistics

Email: bperatt@winona.edu

Flagged courses will normally be in the student’s major or minor program. Departments will need to demonstrate to the University Studies Subcommittee that the courses in question merit the flags. All flagged courses must require the relevant basic skills course(s) as prerequisites (e.g., the "College Reading and Writing" Basic Skill course is a prerequisite for Writing Flag courses), although departments and programs may require additional prerequisites for flagged courses. The University Studies Subcommittee recognizes that it cannot veto department designation of flagged courses.

Critical Analysis courses in the University Studies program are devoted to teaching critical thinking or analytic problem-solving skills. These skills include the ability to identify sound arguments and distinguish them from fallacious ones. The objective of these courses is to develop students’ abilities to effectively use the process of critical analysis. Disciplinary examples should be selected to support the development of critical analysis skills.

These courses must include requirements and learning activities that promote students’ abilities to:

a. Evaluate the validity and reliability of information;

There are essentially two types of reasoning, deductive and inductive. Inductive reasoning is that process by which we as scientists uncover and describe patterns in the world around us via careful observation. Conclusions drawn from inductive reasoning cannot be proven but only verified by repeated observation. It is, however, these principles gained through inductive reasoning that form our axioms for mathematical models of the world. Once those axioms are established, the full scope of their logical ramifications is usually not transparent. Here, deductive reasoning is employed to determine what other conclusions must logically follow if our axioms are taken as valid.

 

Deductive reasoning is the very heart of the mathematical disciplines, and evaluating the validity and reliability of information with respect to various sets of axioms is what mathematicians do. Hence, students in this mathematics course will make heavy use of formal logic to evaluate the validity and reliability of information which they receive either in the form of data or claims about the various mathematical constructs that we are studying.

 

b. analyze modes of thought, expressive works, arguments, explanations, or theories;

The introduction of chaos theory has sparked questions of both a philosophical and mathematical nature. Whereas quantum theory brought the issues of determinism versus random phenomena to the forefront of popular scientific thought, chaos theory has added a wrinkle by uncovering deterministic systems that nevertheless closely mimic random processes. A heavy component of this course will involve the study of nonlinear dynamical systems and the theories concerning their behavior and their validity as useful and adequate models of real-world phenomena.

 

c. recognize possible inadequacies or biases in the evidence given to support arguments or conclusions; and

While a lesser component of the course, students will nevertheless examine the inadequacies of various mathematical models. For example, Euclidean models of the geometry of nature are woefully inadequate in some ways, and fractal geometry claims to more adequately model many phenomena in nature. How well does it fulfill the claims of its proponents? Closely related to fractal geometry is the study of complexity theory in dynamical systems. Many claim that complex systems displaying self-organized criticality, not chaotic systems, are truly a more adequate representation of physical phenomena ranging from heart arrhythmia to earthquake frequency and distribution to the "punctuated equilibrium" phenomenon observed in the evolutionary fossil record.

 

d. advance and support claims.

Finally, a large component of any mathematics course involves the ability to advance and support claims. Such claims must not only be drawn from careful observation of the mathematical models being studied, but they must also be supported by a logical and precise deductive argument which finds its grounding in accepted axiomatic claims.

 

 

 

Winona State University

Course Outline—M315¹

Catalog Description: An introduction to chaos theory and fractal geometry. Topics will include bifurcations, Julia sets, the Mandelbrot set, fractal geometry, iterated function systems and a survey of the applications of this theory to a variety of disciplines.

Major Focus and Objectives: The major focus of this course will be to enable the students to develop an accurate and thorough understanding of this exciting new field of mathematical inquiry and its relationship to his or her chosen discipline. Emphasis will be on discrete systems, though continuous systems will be discussed.

Basic Instructional Methods Utilized: Lecture, discussion, and/or group work on projects will be used.

Course Requirements: Students will be evaluated on their performance on homework assignments, computer assignments, projects, and/or exams.

· A First Course in Chaotic Dynamical Systems: Theory and Experiment, Robert L. Devaney, 1991.

· Encounters with Chaos, Denny Gulick, McGraw Hill, 1992.

· Chaos: An Introduction to Dynamical Systems, Alligood, Sauer, and Yorke, Springer-Verlag, 1997.

· Understanding Nonlinear Dynamics, Kaplan and Glass, Springer-Verlag, 1995.

 

· The Beauty of Fractals: Images of Complex Dynamical Systems, Peitgen and Richter, Springer-Verlag, 1986—An intriguing and comprehensive look at complex Julia Sets and Mandelbrot Sets. Many graphics make this an excellent introduction to complex dynamical systems.

· Chaos in Discrete Dynamical Systems, Abraham, Gardini, and Mira, Springer-Verlag, 1997—A visual introduction to discrete dynamical systems, with a companion CD.

· Dynamical Systems, Clark Robinson, CRC, 1994—A graduate level text on the subject. Fairly comprehensive and well-written from a pure mathematician’s standpoint.

· Encounters with Chaos, Gulick, McGraw Hill, 1992—A short introduction to the main topics of concern in chaos theory. Covers 1-D and 2-D discrete and continuous systems, fractals, complex dynamics, some applications.

· Fractal Designer 3.13, Software—A freeware DOS program which has an interactive user interface for defining iterated function systems graphically.

· The Fractal Geometry of Nature, Mandelbrot, W.H. Freeman and Company, 1984—Benoit Mandelbrot’s manifesto about his work in fractal geometry. In many ways, Mandelbrot was instrumental in bringing together the work of many different mathematicians throughout the twentieth century.

· Fractals Everywhere, Barnsley, Kaufmann Publishers, 2000—Excellent coverage of the mathematics behind fractals. The text is multi-level and has something for everyone. No chaos per se, only fractals.

· Fractals for the Classroom, Peitgen, Jurgens, and Saupe, Springer-Verlag, 1991—A two-volume set which discusses fractals and chaos. One can purchase two lab manuals as well. Much of the material is suitable for use in high school classrooms.

· FractInt, Software, Continually Updated—An excellent freeware DOS program with sophisticated built-in numerical methods for exploring fractals. Also capable of plotting iterated function system fractals with user given data. Available from the web page.

· A First Course in Discrete Dynamical Systems, Holmgren, Springer-Verlag, 1996—A decent text on the subject. Narrow focus.

· Geometer’s Sketchpad, Software, Visual Geometry Project, Key Curriculum Press, 1995—An interactive compass and straightedge program. Incredibly powerful and easy to use. Not designed specifically for fractal construction but has the capability to remember via "scripts." Hence, recursive applications of geometric constructions is fast and relatively easy.

· How Nature Works, Per Bak, Copernicus Books, 1999—Perhaps the most arrogant title for a book that I’ve ever seen, but this book is a good introduction to the field of self-organized criticality. The author has a few axes to grind, but in general, this is an enjoyable read.

· Interactive Differential Equations, Software, Addison-Wesley, 1999—A teaching tool. User can explore many different dynamical systems. Excellent demonstration tool.

· Introduction to Applied Nonlinear Dynamical Systems and Chaos, Wiggins, Springer-Verlag, 1996—A graduate level text on the subject. Fairly comprehensive and well-written from an applied mathematician’s standpoint.

· An Introduction to Chaotic Dynamical Systems, Devaney, Addison-Wesley, 1990—A graduate level version to the above text. Recommended for graduate students in mathematics or experts in other fields who wish to have a rigorous understanding of chaos theory. Only covers discrete systems. Few applications. A nice topological flavor to it.

· Nonlinear Dynamics and Chaos, Strogatz, Perseus Books, 2000—Apparently an excellent introduction to continuous dynamical systems. Lots of applications. Assumes the reader is familiarity with multi-variable calculus and differential equations. Has been described as "marginally less impenetrable than other textbooks on the subject."

· Nonlinear Dynamics and Chaos, Thompson and Stewart, Wiley, 1991—A graduate text for engineers. Authors sometimes don’t use consistent notation, but generally an excellent introduction to the subject for engineering types.

· Nonlinear IFS Editor, Software—A freeware DOS program for rendering non-linear IFS’s. It is, however, written in German.

· Numerical Explorations of Chaotic Systems using DYNAMICS, Software, Nusse and Yorke, Springer-Verlag, 1997—A computer program and lab exercises. DYNAMICS is perhaps the most sophisticated software package available to study chaotic systems. The algorithms it uses were developed at the Institute for Physical Science and Technology at the University of Maryland and are unparalleled in their accuracy and speed. It is available on UNIX and PC platforms.

· Visual Geometry Project Materials, Visual Geometry Project, Key Curriculum Press, 1992. This includes the Platonic Solids activity book, video, and manipulative kit; the Stella Octangula activity book, video, and manipulative kit; and the 3-D Symmetry activity book, video and manipulative kit by the makers of Geometer’s SketchPad. These books are filled with hands-on activities in the form of blackline masters and are ideal for middle-school and secondary teachers.

 

University Studies: Critical Analysis

Flagged courses will normally be in the student’s major or minor program. Departments will need to demonstrate to the University Studies Subcommittee that the courses in question merit the flags. All flagged courses must require the relevant basic skills course(s) as prerequisites (e.g., the "College Reading and Writing" Basic Skill course is a prerequisite for Writing Flag courses), although departments and programs may require additional prerequisites for flagged courses. The University Studies Subcommittee recognizes that it cannot veto department designation of flagged courses.

 

Critical Analysis courses in the University Studies program are devoted to teaching critical thinking or analytic problem-solving skills. These skills include the ability to identify sound arguments and distinguish them from fallacious ones. The objective of these courses is to develop students’ abilities to effectively use the process of critical analysis. Disciplinary examples should be selected to support the development of critical analysis skills.

 

These courses must include requirements and learning activities that promote students’ abilities to:

 

a. Evaluate the validity and reliability of information;

b. analyze modes of thought, expressive works, arguments, explanations, or theories;

c. recognize possible inadequacies or biases in the evidence given to support arguments or conclusions; and

d. advance and support claims.

 

Topics below which include such requirements and learning activities are indicated below using lowercase, boldface letters a.-d. corresponding to these requirements.

 

· Randomness vs. Determinism as it relates to the Natural and Social Sciences. b., c., d.

· Discrete vs. Continuous Dynamical Systems a., d.

· 1-D Discrete Systems

- Basic elements of a discrete system. a.

- Linear and affine systems. a., b.

- Attracting and repelling periodic points. a., b., d.

- Cobweb diagrams. a., d.

- Basins of attraction. a., d.

- Graphing in parameter space. a.

- Bifurcations. a., b., d.

- Period-doubling route to chaos and the Feigenbaum constant. a., b., d.

- At what point does a complicated system become chaotic? Various definitions of chaos. a., b., c., d.

- Topological conjugacy and Cantor sets. a., b., d.

· 2-D Discrete Systems

- A brief introduction to matrices and eigenspaces. a.

- Maps of the plane: sinks, sources, saddles, Henon map, Tinkerbell attractor. a.

- Complex dynamical systems: the Mandelbrot set, Julia sets, Newton's Method. a., b., d.

- Fractal Geometry. a., b., d.

- Iterated Function Systems. a., b.

- Chaos vs. Complexity. a., b., c.

- 1/f-noise in music, earthquakes, etc. a.

· Continuous Dynamical Systems

- Introduction to phase planes for 1-D and 2-D systems. a.

- Linear continuous dynamical systems. a.

- The linearization of nonlinear dynamical systems and its limitations. a., b., d.

- Bifurcations: Hopf, pitchfork, saddle-node, transcritical. a., b.

- Bifurcations and the Tacoma Narrows bridge disaster. b.

- Some canonical systems: unforced pendulum, Lotka-Volterra competing species model, Duffing oscillator, Lorenz system and chaotic regimes, n-body problem, the forced-damped pendulum, and Basins of Wada. b.

- Targeting and Controlling Chaos: OGY method, earthquake-proof buildings, NASA’s utilization of Chaos Theory. b.

 

Approval/Disapproval Recommendations

 

 

Department Recommendation: Approved Disapproved Date

Chairperson Signature Date

Dean’s Recommendation: Approved Disapproved Date

Dean’s Signature Date

*In the case of a Dean’s recommendation to disapprove a proposal, a written rationale for the recommendation

USS Recommendation: Approved Disapproved Date

University Studies Director’s Signature Date

A2C2 Recommendation: Approved Disapproved Date

A2C2 Chairperson Signature Date

Faculty Senate Recommendation: Approved Disapproved Date

FA President’s Signature Date

Academic VP’s Recommendation: Approved Disapproved Date

VP’s Signature Date

President’s Decision: Approved Disapproved Date

President’s Signature Date

Prepared by Barry A. Peratt on March 20, 2001.