615.765.81 - Chaos and Its Applications

Course Structure

The course materials are divided into modules. Clicking “Modules” on the left menu can access the modules. You can also access the modules from the "Home" link on the left.  A module will have several sections including the overview, content, readings, discussions, and assignments. You are encouraged to preview all sections of the module before starting. Typically, a module runs for a period of seven (7) days. See “Course Outline” for more details. Students should regularly check the Announcements for the newest information about the course.

Course Topics

• Review of First Order systems
• Bifurcations
• Flows on Circles
• Introduction to Linear Systems of Differential Equations
• Non-Linear Systems of D.E. 
• Limit Cycles
• Bifurcations of Cycles
• Lorenz System
• Discrete Maps
• Fractals

Course Goals

The main goal of this course is to provide the student with background and mathematical methods which are essential to the solution of single, and systems of differential equations arising in Physics, Biology, and Chemistry. The student will demonstrate understanding of methods covered in the course by solving problems assigned as homework and given on exams, and by presenting a paper illustrating a concrete application using some mathematical method taught in the course.

Course Learning Outcomes (CLOs)

• Deduce the geometry, stability, long-term behavior, and discern different types of bifurcations of solutions to one-dimensional systems.
• Comment on the behavior of solutions of non-linear systems of differential equations, including conservative, and reversible systems, near critical points using linearization and index point theory.
• Show the existence of limit cycles and realize its importance, via the Poincare-Bendixson Theorem, to showing the impossibility of chaos in two-dimensional systems
• Recognize different types of bifurcations in two-dimensional systems, and follow the analysis of the most-famous chaotic three-dimensional system, the Lorenz system, for certain sets of parameter values.
• Deduce the stability of fixed points of discrete maps, and follow the analysis of the dynamics of the logistic map.
• Compute the various dimensions of fractals.
• Create a technical written and oral presentation conveying concepts explored in this course through an application.

Textbooks

Strogatz, Steven: Nonlinear Dynamics and Chaos: Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering, 2nd Edition, CRC Press

ISBN-10: 0813349109

ISBN-13: 978-0813349109

Textbook information for this course is available online through the appropriate bookstore website: For online courses, search the MBS website at http://ep.jhu.edu/bookstore.

Note Any form of the book is acceptable: Electronic form, hard copy new or used, international editions, etc. A student with an earlier edition of the book need not purchase the 2nd edition. This is because the 2nd edition has not been altered enough from the first edition.  In addition, each homework set is typed separately so that there is no need to consult the textbook for the homework.

Required Software

• Matlab

• Pplane (available in the "Additional Materials" folder in the top part of the "Home" page.)  You must have JRE for your Windows machine or Java for your Apple.  For the former, all you need to do is double click on the .jar file.  For the later, open a terminal and type "java -jar pplane.jar".  If you do not want to deal with java altogether, you may also download pplane8 for Matlab which has the same capabilities.

Student Coursework Requirements

It is expected that each module will take approximately 9–13 hours per week to complete. Here is an approximate breakdown: reading the assigned sections of the texts (approximately 2.5–3.5 hours per week) as well as some outside reading, listening to the audio annotated slide presentations (approximately 1–2 hours per week), and writing assignments (approximately 5.5–7.5 hours per week).

This course will consist of four basic student requirements:

• Preparation, Participation, and Discussion Activities (7% of Final Grade Calculation) You are responsible for carefully reading all assigned material and being prepared for discussion. The discussions are mostly on presentations of short applications by teams of students. The applications are based on the course notes and the course text. There are multiple discussion activities to be presented by teams. Each team consists of two students randomly selected by me. If the course has an odd number of students, then one team will consist of three students.  Details of these activities are in the Modules and Announcements.

The discussions typically run for a period of one to two weeks. Posting your team’s presentation of application is part one of your grade for module discussions (i.e., Timeliness).

Part two of your grade for module discussion is your interaction: responding to classmates’ or teams’ postings with thoughtful responses (i.e., Critical Thinking). Be concise and clear in your postings and in your responses to your classmates' postings. Feel free to agree or disagree with your classmates. Please ensure that your postings are civil and constructive.

There are also two special discussion activities one at the beginning of the semester where you are asked to briefly introduce yourselves and one at the end of classes after the project presentations are recorded and made available to the class. Each student then is required to watch the recordings and comment on the project presentations or ask questions, if there are any questions.

I will monitor module discussions and will respond to some of the discussions as discussions are posted.  Evaluation of preparation and participation is based on contribution to discussions.

Preparation and participation is evaluated by the following grading elements: Timeliness (50%), Student Interaction and Critical Thinking (50%)

Preparation and participation is graded as follows:

100–90, A—range: Timeliness [regularly participates; all required postings; early in discussion; throughout the discussion]; Critical Thinking [correct thoughts and analysis].

89.99–80, B—range: Timeliness [frequently participates; all required postings; some not in time for others to read and respond]; Critical Thinking [mostly correct thoughts and analysis].

79.99–70, C—range: Timeliness [infrequently participates; all required postings; most at the last minute without allowing for response time]; Critical Thinking [generally correct; some analysis].

• Assignments (30% of Final Grade Calculation)

There are 12 assignments that consist of mathematical problems related to the material covered in the course. The assignments are the very important. They carry the most single item weight of your grade. They also point towards most of the types of questions in the exams. Please work diligently to submit very good assignment solutions.

When you submit your assignment for grading please:

1. Include your name and assignment identifier.
2. Submit in Canvas only one file in PDF format. The best way to achieve this is to use a scanner to scan your hard copy into one PDF file.
3. Avoid uploading one page at a time.
4. Your uploaded file name should be as follows: “YOUR LAST NAME_ASSIGNMENT_NUMBER”
5. Avoid emailing your assignments, unless it is absolutely necessary (for example, if Canvas would not upload your document).
6. Avoid uploading pictures of your assignment because they create big files that are hard to upload, download, or save.
7. If you need to insert extra pages please include your name and an extra page number indicator (i.e., extra page x of y).
8. Each problem should have the assumptions, computations, and conclusions/discussion delineated.
9. Your writing should be clear.
10. The computations should be presented in logical order and sequentially.

All assignments are due according to the dates in Modules and Course Outline.

With the exception of Assignments 6 and 13, late submissions will be accepted up to a maximum of three days after the day they are due. The grade for late submission is reduced by 10% for each day that the assignment is overdue (no exceptions without prior coordination with me). The only exception are Assignments 6 and 13: the due date is firm and no extensions will be allowed. The reason is that I would like to make the key to this assignment available as soon as possible so that you can prepare for the midterm and final exams.

Note: The lowest assignment grade will be dropped.

Each assignment consists of several problems, usually similar to the examples presented in the course. Each problem carries an indicated number of points.

Each problem is graded as follows:

1. The method is correct (50%)
2. The computation, graphs, if any, and final answer are correct (40%)
3. The presentation is clear and logical (10%)

• Course (Team) Project (18% of Final Grade Calculation)

The course includes a mandatory project. The project consists of a written paper and a short presentation in Adobe Connect. You are to work in teams of two. Each team submits one paper.  Each student participates in the short presentation. If the course has an odd number of students, then one team will consist of three students. I prefer that you pick your partners for the project. If you cannot decide on a partner, then you will be assigned one by me.

1. Correctness: correct equations, proofs, graphs, & conclusions (40%)
2. Amount of work: strong evidence that the work is not superficial, routine, or trivial (30%)
3. Writing style: clear and concise writing with appropriate graphs, illustrations, etc. (10%)
4. Project presentation and clarity of presentation (20%)

It is also understood that each team member participates equally in the work involved to produce the project. I reserve the right to lower the project grade of a student for whom there is evidence that he/she did not fully participate in all the work needed for the project.

• Exams (45% of Final Grade Calculation, combined from 22.5% for Midterm and 22.5% for Final)

The midterm exam will be available in Module 7 and the final exam will be available in Module 14.  The dates and times of the exams are in the Course Outline document. Also, you will be reminded, via an Announcement in Canvas, about the time interval when the exam will be available. Students may use the course notes, the course text, and their own notes to complete the exams. If you are unable to take the exam within the allocated time interval, please contact me to make some other arrangement.

1. The method is correct (50%)
2. The computation, graphs, if any, and final answer are correct (40%)
3. The presentation is clear and logical (10%)

Grading Policy

Student assignments are due according to the dates in the Calendar and Assignments items in the corresponding modules. I will post grades one week after assignment due dates.

A grade in the A range indicates achievement of consistent excellence and distinction throughout the course—that is, conspicuous excellence in all aspects of assignments and discussion in every week.

A grade in the B range indicates work that meets all course requirements on a level appropriate for graduate academic work. These criteria apply to both undergraduates and graduate students taking the course. A grade below B− does not count towards graduate credit.

Course Evaluation

Final grades will be determined by the following weighting:

Course Policies

The policies on collaborations and discussions are:

Assignments

You are allowed to discuss concepts and ideas of solutions of assignment problems. However, the final submitted homework should be entirely your own work. Copying the work of another will receive a warning at the first incident. Any further incidents will result in receiving a zero on the assignment and the matter will be referred to the Associate Dean.

Exams

The work on the midterm exam and the final exam should be absolutely and entirely your own work.  There should be no discussion or assistance from any source on the exam questions. Copying the work of another individual or receiving assistance of any form will result in receiving a zero on the exam and the matter will be referred to the Associate Dean.

Project

The project does not have to be original or research work. However, the writing of the paper should be your own work. If you are using some published material, you need to provide a complete citation of your source(s).

Student collaboration on the project is highly encouraged. Team members should fully collaborate to achieve a quality product. Different teams are allowed to discuss concepts of their projects with other teams. However, teams are not allowed to copy material from each other.

Plagiarism

Plagiarism is defined as taking the words, ideas or thoughts of another and representing them as one's own. If you use the ideas of another, provide a complete citation in the source work; if you use the words of another, present the words in the correct quotation notation (indentation or enclosed in quotation marks, as appropriate) and include a complete citation to the source. See the course text for examples.

Academic Policies

Deadlines for Adding, Dropping and Withdrawing from Courses

Students may add a course up to one week after the start of the term for that particular course. Students may drop courses according to the drop deadlines outlined in the EP academic calendar (https://ep.jhu.edu/student-services/academic-calendar/). Between the 6th week of the class and prior to the final withdrawal deadline, a student may withdraw from a course with a W on their academic record. A record of the course will remain on the academic record with a W appearing in the grade column to indicate that the student registered and withdrew from the course.

Academic Misconduct Policy

All students are required to read, know, and comply with the Johns Hopkins University Krieger School of Arts and Sciences (KSAS) / Whiting School of Engineering (WSE) Procedures for Handling Allegations of Misconduct by Full-Time and Part-Time Graduate Students.

This policy prohibits academic misconduct, including but not limited to the following: cheating or facilitating cheating; plagiarism; reuse of assignments; unauthorized collaboration; alteration of graded assignments; and unfair competition. Course materials (old assignments, texts, or examinations, etc.) should not be shared unless authorized by the course instructor. Any questions related to this policy should be directed to EP’s academic integrity officer at ep-academic-integrity@jhu.edu.

Students with Disabilities - Accommodations and Accessibility

Johns Hopkins University values diversity and inclusion. We are committed to providing welcoming, equitable, and accessible educational experiences for all students. Students with disabilities (including those with psychological conditions, medical conditions and temporary disabilities) can request accommodations for this course by providing an Accommodation Letter issued by Student Disability Services (SDS). Please request accommodations for this course as early as possible to provide time for effective communication and arrangements.

For further information or to start the process of requesting accommodations, please contact Student Disability Services at Engineering for Professionals, ep-disability-svcs@jhu.edu.

Student Conduct Code

The fundamental purpose of the JHU regulation of student conduct is to promote and to protect the health, safety, welfare, property, and rights of all members of the University community as well as to promote the orderly operation of the University and to safeguard its property and facilities. As members of the University community, students accept certain responsibilities which support the educational mission and create an environment in which all students are afforded the same opportunity to succeed academically. 

For a full description of the code please visit the following website: https://studentaffairs.jhu.edu/policies-guidelines/student-code/

Classroom Climate

JHU is committed to creating a classroom environment that values the diversity of experiences and perspectives that all students bring. Everyone has the right to be treated with dignity and respect. Fostering an inclusive climate is important. Research and experience show that students who interact with peers who are different from themselves learn new things and experience tangible educational outcomes. At no time in this learning process should someone be singled out or treated unequally on the basis of any seen or unseen part of their identity. 
 
If you have concerns in this course about harassment, discrimination, or any unequal treatment, or if you seek accommodations or resources, please reach out to the course instructor directly. Reporting will never impact your course grade. You may also share concerns with your program chair, the Assistant Dean for Diversity and Inclusion, or the Office of Institutional Equity. In handling reports, people will protect your privacy as much as possible, but faculty and staff are required to officially report information for some cases (e.g. sexual harassment).

Course Auditing

When a student enrolls in an EP course with “audit” status, the student must reach an understanding with the instructor as to what is required to earn the “audit.” If the student does not meet those expectations, the instructor must notify the EP Registration Team [EP-Registration@exchange.johnshopkins.edu] in order for the student to be retroactively dropped or withdrawn from the course (depending on when the "audit" was requested and in accordance with EP registration deadlines). All lecture content will remain accessible to auditing students, but access to all other course material is left to the discretion of the instructor.