Math 538 — Discrete Dynamical Systems and Chaos — Syllabus

Some details of the syllabus may change between semesters; consult your professor's syllabus!

Stable Course Components

The course components described in this section are mostly professor independent, and should not greatly change from semester-to-semester.

Catalog Description

One- and two-dimensional iterated maps, equilibria and their stability, sensitive dependence on initial conditions, Lyapunov exponents, horseshoe maps, period doubling, chaotic attractors, Poincare maps, stable/unstable manifolds, bifurcations. Applications in biology, chemistry, physics, engineering, and other sciences.

Prerequisites

1. Math 237 (Elementary Differential Equations),
2. Math 330 (Advanced Calculus I),
3. Math 340 (Mathematical Programming), and
4. Math 342B (Methods of Applied Mathematics II).

Overview

Dynamical Systems is the study of phenomena that evolve in space and/or time. Whether a particular system comes from Biology, Physics, Chemistry, or even the Social Sciences, Dynamical Systems is the subject that provides the mathematical tools for its analysis. The seminal work by Lorenz in 1963, gave scientists insight to recognize a new type of motion called Chaos. Chaotic systems, which can be very simple, are capable of generating erratic behavior that is different from the one produced by quasi-periodic systems with a large number of frequencies of oscillations. Chaotic systems and their applications are the subjects of this course. The course is intended for senior undergraduate and first year graduate students in Applied Mathematics, Computational Science, Engineering, Physics, Chemistry, Biology, etc. Examples from interdisciplinary areas will be covered. Most of the concepts and examples will be supplemented with Matlab-based codes. As part of the course, students will be given access to a computer laboratory to complete the computer-based coursework.

Student Learning Objectives

• Visualize the dynamics of discrete maps using cobweb plots. Find and analyze fixed points for stability. Identify periodic orbit and basins of attraction of attraction for fixed points and periodic orbits.
• Characterize chaotic motion using Poincare sections and Lyapunov Exponents.
• Recognize that a geometrical object can have fractal dimension. Characterize fractals using the concept of fractal dimension.
• Extend the above ideas and concepts to two-dimensional maps. Identify the different types of bifurcations for two-dimensional maps (saddle-node, period-doubling, transcritical).
• Deploy the one or several concepts learned in class towards an independent research project. Usually in small groups of 2-3 students during the last 4-5 weeks of the semester.

General Policies and Information

The information in this section applies to all courses offered by the department

Students with Disabilities

If you are a student with a disability and believe you will need accommodations for this class, it is your responsibility to contact the Student Ability Success Center at (619) 594-6473. To avoid any delay in the receipt of your accommodations, you should contact Student Ability Success Center as soon as possible. Please note that accommodations are not retroactive, and that I cannot provide accommodations based upon disability until I have received an accommodation letter from Student Ability Success Center. Your cooperation is appreciated.

Student Privacy and Intellectual Property

The Family Educational Rights and Privacy Act (FERPA) mandates the protection of student information, including contact information, grades, and graded assignments. I will not post grades or leave graded assignments in public places. Students will be notified at the time of an assignment if copies of student work will be retained beyond the end of the semester or used as examples for future students or the wider public. Students maintain intellectual property rights to work products they create as part of this course unless they are formally notified otherwise.

Mathematics and Statistics Learning Center

The SDSU Math & Stat Learning Center is in the Love Library, Room LL-328. "The Math and Stats Learning Center is open to support students in all lower division math courses at SDSU. We have tutors available for walk-in help during all open hours. TAs for Math 141, 150, 151, and 252 also hold their office hours there. Please see the schedule of when the TAs for your class will be in the center by going to our website: mlc.sdsu.edu. The MLC is supported by your student success fee. We strongly encourage you to use this wonderful, free resource. Some students believe that they should not need to ask for help. But, research has shown that the average grade for students who attend the MLC is one half grade higher than those who don’t seek such support."

If you are enrolled in a class which does not have targeted support, the MLC can still serve as a great math study/meeting place; and if you are interested in becoming a tutor in the center, keep an eye on the center's webpage for hiring announcements.

Cheating and Plagiarism

Students are generally encouraged to study together, and to work together to solve exercises. Finals, Midterms, Quizzes, Project, and other designated "individual work" activities must be completed without assistance. All violations will be reported to the Center for Student Rights and Responsibilities and will also result in score/grade reductions at the professor's discretion. Please review SDSU's full policy on academic honesty.

Examples of academic dishonesty include but are not limited to

• copying, in part or in whole, from another's test or other examination
• obtaining copies of a test, an examination, or other course material without the permission of the instructor
• collaborating with another or others in work to be presented without the permission of the instructor
• falsifying records, laboratory work, or other course data
• submitting work previously presented in another course, if contrary to the rules of the course
• altering or interfering with grading procedures
• assisting another student in any of the above
• using sources verbatim or paraphrasing without giving proper attribution (this can include phrases, sentences, paragraphs and/or pages of work)
• copying and pasting work from an online or offline source directly and calling it your own
• using information you find from an online or offline source without giving the author credit
• replacing words or phrases from another source and inserting your own words or phrases

Religious Observances

According to the University Policy File, students should notify the instructors of affected courses of planned absences for religious observances by the end of the second week of classes.

Medical-Related Absences

Students are instructed to contact their professor/instructor in the event they need to miss class, etc. due to an illness, injury or emergency. All decisions about the impact of an absence, as well as any arrangements for making up work, rest with the instructors. Student Health Services (SHS) does not provide medical excuses for short-term absences due to illness or injury. When a medical-related absence persists beyond five days, SHS will work with students to provide appropriate documentation. When a student is hospitalized or has a serious, ongoing illness or injury, SHS will, at the student's request and with the student’s consent, communicate with the student’s instructors via the Vice President for Student Affairs and may communicate with the student’s Assistant Dean and/or the Student Ability Success Center