Course Syllabus - Nonlinear Dynamics and Chaos

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Required Texts: Chaos: An Introduction to Dynamical Systems, by Alligood, Sauer and Yorke, 2000.

We will cover the material in Chapters 1-9 of the text. Additional topics may be covered as time permits. You will also be assigned additional readings from other sources and projects during this course. It is the intent that this course will be fun, interesting and shed light on some current topics in applied mathematics.

What is Chaos?

The idea of dynamical chaos was first glimpsed by Poincaré when he entered a contest sponsored by the king of Sweden. One of the questions in this contest was to show rigorously that the solar system as modeled by Newton's equations is dynamically stable. The question was nothing more than a generalization of the famous three body problem, which was considered one of the most difficult problems in mathematical physics. In essence, the three body problem consists of nine simultaneous differential equations (all linear, each of second order). The difficulty was in showing that the general solution converges since any solution will be given in terms of infinite series. While Poincaré did not succeed in giving a complete solution, he made such a major headway in attacking the problem that he was awarded the prize anyway. The distinguished Weierstrass, who was one of the judges, said, "this work cannot indeed be considered as furnishing the complete solution of the question proposed, but that it is nevertheless of such importance that its publication will inaugurate a new era in the history of celestial mechanics."

To show how visionary Poincaré was, it is perhaps best if he described the Hallmark of Chaos - sensitive dependence on initial conditions - in his own words:

If we knew exactly the laws of nature and the situation of the universe at the initial moment, we could predict exactly the situation of that same universe at a succeeding moment. but even if it were the case that the natural laws had no longer any secret for us, we could still only know the initial situation approximately. If that enabled us to predict the succeeding situation with the same approximation, that is all we require, and we should say that the phenomenon had been predicted, that it is governed by laws. But it is not always so; it may happen that small differences in the initial conditions produce very great ones in the final phenomena. A small error in the former will produce an enormous error in the latter. Prediction becomes impossible, and we have the fortuitous phenomenon. - in a 1903 essay "Science and Method"

Student Learning Outcomes

Upon completion of this course, students will be able to

• Find fixed points and periodic orbits of iterated one-dimensional mappings and discuss their stability;
• Use cobweb diagrams to illustrate stability of a fixed point or periodic orbit;
• Identify commonly encountered local and global bifurcations;
• Connect Lyapunov exponents to chaos and compute them for simple examples;
• Construct and analyze a wide range of fractals;
• Understand and compute various fractal dimensions;
• Find and identify features of Mandelbrot and Julia sets;
• Understand and create Iterated Function Systems;
• Determine fixed points, the linearization about the fixed points, and their stability for one- and two-dimensional systems of ODEs;
• Understand the distinction between hyperbolic and non-hyperbolic fixed points;
• Sketch a phase portraits of linear and nonlinear one- and two-dimensional systems of ODEs;
• How to identify features of nonlinear systems such as periodicity, limit cycles, and strange attractors;
• Use the Poincaré-Bendixson theorem and polar coordinates to investigate the existence of limit cycles;
• Investigate bifurcations of one-dimensional dynamical systems in general; draw bifurcation diagrams;
• Apply standard nonlinear techniques in applications from different disciplines;

Course Requirements: