## Texts

There are no required texts for this course. The following books may be helpful to supplement lectures and the book draft.

- David P. Feldman. Chaos and Fractals: An Elementary Introduction. Oxford University Press. 2012.
- Daniel Kaplan and Leon Glass. Understanding Nonlinear Dynamics. Birkhauser. 1995.
- Steven Strogatz. Nonlinear Dynamics and Chaos. Westview Press. 1994.

## Assignments

- Assignment One
- Assignment Two
- Assignment Three
- Assignment Four
- Assignment Five
- Assignment Six
- Assignment Seven
- Assignment Eight
- Assignment Nine
- Assignment Ten
- Assignment Eleven
- Assignment Twelve

## Code

- Iterating functions
- Another version of iteration program
- Euler code
- Basic Lotka Volterra code
- Fancy Lotka Volterra code
- Histograms for the logistic equation

## Lecture Notes/Book Draft

## Course Structure

This tutorial will meet two to three times a week. Class meetings will be a combination of lecture, discussion, group work, and programming sessions. For almost all class meetings I will assign exercises that students will do in groups and then discuss in class. Students will do a final project in which they look at a particular application of dynamical systems to an area of science. Using this analysis as a starting point, students will also consider what general lessons from dynamical systems are important for this scientific area.## Outline

**Introduction I: Dynamical Systems**Finite difference equations and differential equations. Solution methods, focusing almost exclusively on numerical methods.**Introduction II: Global and Qualitative Analysis of Dynamical Systems.**Global analysis of dynamical systems. Stable and unstable solutions. Fixed points and periodic behavior. Phase space.**Small Changes Yield Big Changes.**Bifurcations. Catastrophes. Hystersis. Multiple equilibria.**Disorder from Order.**Chaos. Sensitive dependence on initial conditions (butterfly effect). Lyapunov exponents.**Coexistence of Order and Disorder.**Stable chaos. Invariant densities. Strange attractors.**Complex Behavior from Simple Rules.**Strange attractors. Pattern formation. Self-organization and emergence.**Conclusion.**

## Links

- A particularly good matplotlib tutorial
- A good tutorial on histograms

This course is a survey of dynamical systems, the field of applied mathematics that studies systems that change over time. The modern study of dynamical systems includes examining particular systems or areas of application, as well as looking at systems more broadly and abstractly to develop generally applicable tools for studying dynamical systems or to classify different sorts of behavior.

This course is intended for motivated students with strong math backgrounds who wish to gain an overview of dynamical systems and to discuss and debate the insights the study of dynamical systems holds for the physical, natural, and social sciences. Using both differential equations and difference equations as our main items of study, we will cover standard topics in dynamical systems, including phase space, bifurcation diagrams, chaotic behavior, sensitive dependence on initial conditions, strange attractors, embedding and attractor reconstruction, and Lyapunov exponents. A central theme that emerges from the study of dynamical systems is that there is a subtle relationship between order and disorder. Unpredictable behavior can arise from deterministic dynamical systems, and complex behavior can have simple origins. We shall see that predictability and unpredictability, simplicity and complexity, and order and disorder are not opposites, but are often exist simultaneously in the same dynamical system.

Evaluation will be based on participation in seminar-style class sessions, problems sets, and a final project and presentation.

Intermediate. No lab fee. Prerequisites: Calculus II and permission of instructor. Experience writing simple computer programs (in any language) will be helpful, but not required. Class size limited to 5.