This course examines ordinary differential equations from a geometric point of view and involves significant use of phase portrait diagrams and associated concepts, including equilibrium points, orbits, limit cycles, and domains of attraction. Various methods are discussed to determine existence and stability of equilibrium points and closed orbits. Methods are discussed for analyzing nonlinear differential equations (e.g., linearization, direct, perturbation, and bifurcation analysis). An introduction to chaos theory and Hamiltonian systems is also presented. The techniques learned will be applied to equations from physics, engineering, biology, ecology, and neural networks (as time permits).
The course materials are divided into 14 modules which can be accessed by clicking Course Modules on the left menu. A module will have several sections including the module-at-a-glance, readings, video lectures and related content, discussions, and quizzes. Students should regularly check the Calendar and Announcements for assignment due dates. Modules begin on Wednesdays and complete on Tuesdays.
1. Develop qualitative and geometric solutions to nonlinear ordinary differential equations.
2. Construct phase-plane diagrams and associated concepts, including equilibrium points, orbits, limit cycles, and domains of attraction as applied to nonlinear differential equations.
3. Relate techniques of nonlinear ordinary differential equations and dynamical systems to problems in engineering, physics, biology, and economics.
4. Illustrate the basic concepts of chaos theory for dynamical systems.
The course uses dynamics to approach the difficult subject of nonlinear ordinary differential equations. The student will learn new, unique skills in this class to use visual, geometric, analytical, and qualitative methods to provide insight into dynamical systems.
Strogatz, Steven H. Nonlinear Dynamics and Chaos with Student Solutions Manual: With Applications to Physics, Biology, Chemistry, and Engineering. CRC Press, 2018.
You do not need to purchase this textbook. A free electronic version is linked into the Canvas course site by selecting the EReserves link on the left menu bar.
Webcam, microphone (if provided with computer, this is usually adequate along with adequate lighting and no background sound interference). Standard MS suite applications recommended. MATLAB or similar may be useful on a few homework applications but this is not required.
It is expected that each module will take approximately 10-12 hours per week to complete. Here is an approximate breakdown: reading the assigned sections of the texts (approximately 2-3 hours per week), as well as some videos (1-2 hours) and assignments (approximately 5–6 hours per week).
This course will consist of five basic student requirements:
1. Assignments (40% of final grade)
Weekly assignments are from the Stogatz textbook. They are an opportunity for you to practice the content for each week. Instructor monitors and provides guidance as necessary. In addition, instructor is available via email or weekly synchronous office hours.
2. Discussion board assignments (10% of final grade)
Weekly discussion board assignments require you to think about real-world applications for the mathematical concepts you will be learning. You are required to respond to each discussion prompt as well as respond to at least two other students’ post.
3. Quizzes (30% of final grade)
There are three quizzes in the course, in modules 5, 8, and 11. Quizzes cover material from previous weeks that is practiced in the weekly assignments.
4. Final Exam (20% of final grade)
The final exam is in module 14. You have all week to complete the exam.
Assignments are due according to the dates posted in the course site. Students may check these due dates in the Course Calendar or the Assignments in the corresponding modules. Grades will post no later than one week after assignment due dates. A grade of A 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 of B 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.
The following grades are used for this course: A+, A, A– (excellent), B+, B, B– (good), C (unsatisfactory), F (failure), I (incomplete). A grade of F indicates the student’s failure to complete or comprehend the course work.
A course for which an unsatisfactory grade (C or F) has been received may be retaken. The original grade is replaced with an R. If the failed course includes laboratory, both the lecture and laboratory work must be retaken unless the instructor indicates otherwise. A grade of W is issued to those who have dropped the course after the refund period but before the drop deadline. The transcript is part of the student’s permanent record at the university. No grade may be changed except to correct an error, to replace an incomplete with a grade, or to replace a grade with an R.
The Whiting School assumes that students possess acceptable written command of the English language. It is proper for faculty to consider writing quality when assigning grades.
For incomplete grades, please see the Graduate Programs catalogue for the Whiting School of Engineering.
The course grading scale is the following:
100–98 = A+
97–94 = A
93–90 = A−
89–87 = B+
86–83 = B
82–80 = B−
79–70 = C
<70 = F
Assignments are due according to the dates posted in the Canvas course site. Students may check these due dates in the Course Calendar or the Assignments in the corresponding modules. Grades will post no later than one week after assignment due dates. A grade of A 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 of B 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.
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 email@example.com.
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, firstname.lastname@example.org.
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/
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).
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.