This first course in nonlinear dynamics and chaos is aimed at upper-level undergraduate and graduate students. We will use analytical methods, concrete examples, and geometric intuition to develop the basic theory of dynamical systems, starting with first-order differential equations and their bifurcations, followed by phase plane analysis, limit cycles, and their bifurcations, and culminating with the Lorenz equations, chaos, iterated maps, period doubling, renormalization, fractals, and strange attractors.
Your grade will be determined by nine homework assignments and a final exam. Homework will comprise seventy percent of your grade, and the final exam will comprise the remaining thirty percent. A combined total of ninety percent or higher will guarantee an A, eighty percent a B, seventy percent a C, and sixty percent a D.
| Required Text | Nonlinear Dynamics and Chaos (Second Edition) by Steven H. Strogatz |
|---|---|
| Class Hours | Mondays, Wednesdays, and Fridays from 11:00am to 11:50am |
| Class Location | Science and Mathematics Learning Center 356 |
| Office Hours | Mondays from 2:00pm to 3:00pm and Wednesdays from 2:00pm to 4:00pm |
| Office Location | Science and Mathematics Learning Center 226 |
You have the option of completing a project to replace your two lowest homework grades. For the project, you need to read a research paper and complete the following tasks:
There is no minimum length, and you may submit an early draft for my feedback if you desire. The final draft will be due at the final exam. You may work as a pair, but the expectations will be higher in this case. Potential topics include:
Each assignment is due in class on the date indicated. For some of the exercises, you will need the PPLANE utility for Matlab, which can be downloaded here for version R2014a or older and here for version R2014b or newer.
| 26 January | Exercises 2.1.1-2.1.3, 2.2.6, 2.2.8, 2.3.2, 2.4.7, 2.5.4, 2.6.1, 2.7.6 (Solutions) |
|---|---|
| 6 February | Exercises 3.1.3, 3.2.4, 3.3.2, 3.4.2, 3.4.12, 3.5.2, 3.5.3, 3.5.8, 3.6.2, 3.6.6 (Solutions) |
| 23 February | Exercises 4.1.1, 4.1.2, 4.2.1, 4.3.3, 4.4.1, 5.1.1, 5.1.2, 5.2.1, 5.2.2, 5.2.8 (Solutions) |
| 20 March | Exercises 6.2.1, 6.2.2, 6.3.1, 6.3.11, 6.3.12, 6.4.7, 6.5.1, 6.5.20, 6.6.1, 6.6.5 (Solutions) |
| 30 March | Exercises 7.1.5, 7.1.8, 7.2.5, 7.2.7, 7.3.1, 7.6.1, 7.6.2, 7.6.3 (Solutions) |
| 10 April | Exercises 8.1.2, 8.1.3, 8.1.4, 8.1.6, 8.2.2, 8.2.3, 8.2.4 (Solutions) |
| 24 April | Exercises 9.2.1, 9.2.2, 9.2.6, 9.3.8, 9.4.2 |
| 1 May | Exercises 10.3.1, 10.3.2, 10.5.1, 11.1.1, 11.2.2, 11.2.3 |
| 12 January | Chapter 1 (Overview of Nonlinear Dynamics and Chaos) |
|---|---|
| 14 | Sections 2.0-2.2 (Introduction to Flows on the Line, Fixed Points and Stability) |
| 16 | Sections 2.4 (Linear Stability Analysis) |
| 19 | Martin Luther King, Jr. Day |
| 21 | Sections 2.5-2.7 (Existence and Uniqueness, Impossibility of Oscillations, Potentials) |
| 23 | Sections 3.0-3.1 (Introduction to Bifurcations, Saddle-Node Bifurcation) |
| 26 | Sections 3.2-3.3 (Transcritical Bifurcation, Laser Threshold) |
| 28 | Section 3.4 (Pitchfork Bifurcation) |
| 30 | Section 3.5 (Overdamped Bead on a Rotating Hoop) |
| 2 February | Section 3.6 (Imperfect Bifurcations and Catastrophes) |
| 4 | Sections 4.0-4.2 (Introduction to Flows on the Circle, Examples and Definitions, Uniform Oscillator) |
| 6 | Section 4.3 (Nonuniform Oscillator) |
| 9 | Sections 4.4-4.5 (Overdamped Pendulum, Fireflies) |
| 11 | Sections 5.0-5.1 (Introduction to Linear Systems, Definitions and Examples) |
| 13 | Section 5.2 (Classification of Linear Systems) |
| 16 | Sections 6.0-6.2 (Introduction to the Phase Plane, Phase Portraits, Existence, Uniqueness, and Topological Consequences) |
| 18 | Section 6.3 (Fixed Points and Linearization) |
| 20 | Section 6.5 (Conservative Systems) |
| 23 | Section 6.6 (Reversible Systems) |
| 25 | Section 6.8 (Index Theory) |
| 27 | Sections 7.0-7.2 (Introduction to Limit Cycles, Examples, Ruling Out Closed Orbits) |
| 2 March | Section 7.3 (Poincaré-Bendixson Theorem) |
| 4 | Sections 7.4-7.5 (Liénard Systems, Relaxation Oscillators) |
| 6 | Section 7.6 (Weakly Nonlinear Oscillators) |
| 9 | Spring Break |
| 11 | Spring Break |
| 13 | Spring Break |
| 16 | Sections 8.0-8.1 (Bifurcations Revisited, Saddle-Node, Transcritical, and Pitchfork Bifurcations) |
| 18 | Section 8.2 (Hopf Bifurcations) |
| 20 | Section 8.3 (Oscillating Chemical Reactions) |
| 23 | Section 8.4 (Global Bifurcations of Cycles) |
| 25 | Section 8.6 (Coupled Oscillators and Quasiperiodicity) |
| 27 | Section 8.7 (Poincaré Maps) |
| 30 | Sections 9.0-9.1 (Introduction to the Lorenz Equations, A Chaotic Waterwheel) |
| 1 April | Section 9.2 (Simple Properties of the Lorenz Equations) |
| 3 | Section 9.3 (Chaos on a Strange Attractor) |
| 6 | Section 9.4 (Lorenz Map) |
| 8 | Sections 10.0-10.1 (Introduction to One-Dimensional Maps, Fixed Points and Cobwebs) |
| 10 | Sections 10.2-10.3 (Numerics and Analysis of the Logistic Map) |
| 13 | Section 10.4 (Periodic Windows) |
| 15 | Section 10.5 (Liapunov Exponent) |
| 17 | Section 10.6 (Universality and Experiments) |
| 20 | Section 10.7 (Renormalization) |
| 22 | Sections 11.0-11.2 (Introduction to Fractals, Countable and Uncountable Sets, Cantor Set) |
| 24 | Sections 11.3-11.4 (Dimension of Self-Similar Fractals, Box Dimension) |
| 27 | Section 11.5 (Pointwise and Correlation Dimensions) |
| 29 | Sections 12.0-12.1 (The Simplest Examples of Strange Attractors) |
| 1 May | Reflection and Review |