350 Dynamics and Chaos Spring 2022 – John McCarthy, Ph.D.

Instructor          John E. M^cCarthy
Class                          TuTh 10.00-11.20, 103 Lopata Hall
Midterm                      Thursday March 10
Final                           Tuesday May 10th, 6.00-8.00 p.m.

Office 105 Cupples I
Office Hours Monday 11.00-11.50, Tuesday 3.00-3.50, Thursday 2.30-3.20

In office and via Zoom: Meeting ID: 932 2571 1839

Prerequisites Math 217 (Differential Equations) and 233 (Multivariable Calculus)

Description
One of the great mathematical discoveries of the 20th century is that nearly all naturally occuring systems are chaotic –
small changes in the initial conditions will lead to large changes in the long-term behavior.
This means that predictions can only be made for a characteristic time period.
This can be days for the weather, hundreds of millions of years for planetary motion, or seconds for some chemical reactions.
But there is a theoretical limit to how well predictions can do.

This course shall investigate dynamical systems, with applications to various fields of science, and how chaos arises.

Homework

There will be weekly homework sets during the semester, assigned on Thursday and due the following Thursday.

Content

Flows on the line: Chapter 2
Bifurcations : Chapter 3
Linear Systems : Chapter 5
Phase Plane : Chapter 6
Limit Cycles : Sections 7.0- 7.3
Bifurcations Revisited: Sections 8.0-8.3
Lorenz equations : Sections 9.2-9.5
Chaos : Section 9.3
Strange attractors : Section 9.3
One dimensional systems: Chapter 10
Fractals: Chapter 11
Strange Attractors: Chapter 12

Basis for Grading

Attendance and class participation will be 10% of your grade, homework will be 30%, the midterm will be 20%, and the final will be 40%.

Homework

Homework is an extremely important part of the course. Whilst talking to other people about it is not dis-allowed, too often this degenerates into one person solving the problem, and other people copying them (often justified to themselves by saying “I provide the ideas, X does the details” – but the details are the key. If you can’t translate the idea into a real proof, you don’t understand the material well enough). So I shall introduce the following rules:
(a) You can only talk to some-one else about a problem if you have made a genuine effort to solve it yourself.
(b) You must write up the solutions on your own. Suspiciously similar write-ups will receive 0 points.

Class

You must come to class every day, and participate in class discussions, unless you are ill or quarantining. In these events, please email me.
I also expect you to stay abreast of the material we are covering, and may call on you at any time to answer a question.

Text Nonlinear dynamics and Chaos, Second Edition, by S.H. Strogatz (CRC 2015).

Other books Chaos: Making a New Science by James Gleick is one of the best general interest science books ever written,
capturing the excitement of the discoveries and describing the people who made them.

Websites Here are some videos: http://www.clausewitz.com/mobile/chaosdemos.htm

COVID-19 Health and Safety Protocols Exceptions to course attendance policies, expectations, and requirements because of a COVID-19 diagnosis, symptoms consistent with COVID-19, or exposure to a person with a confirmed or suspected COVID-19 diagnosis that requires quarantine or isolation will be made in collaboration between the student and instructor. In these cases, please notify your instructor as soon as possible to discuss appropriate accommodations. While on campus, it is imperative that students follow all public health guidelines established to reduce the risk of COVID-19 transmission within our community. The full set of University protocols can be found at https://covid19.wustl.edu/health-safety/. This includes:

• Completing a self-screening using the WashU COVID-19 Screening app every day before coming to campus or leaving your residence hall room. If you do not receive a green check and pass the screening, you are not permitted to come to campus or leave your residence hall room. You must contact the COVID Call Center (314-362-5056) or the Habif Health and Wellness Center (314 935-6666) immediately. Note: In addition to the symptoms listed in the screening tool, everyone also should pay attention to symptoms that are new or different for you, including things like headache and congestion, particularly in combination with diarrhea. These can also be signs of COVID-19. Call the COVID Call Center or Habif to report these symptoms.

• Complying with universal masking. All individuals on campus must wear disposable masks or cloth face coverings while occupying indoor public settings, including: multi-person offices, hallways, stairwells, elevators, meeting rooms, classrooms and restrooms. Masks are encouraged but not required for outdoor activities, particularly at large events or in crowded settings. Students with disabilities for whom masked instructors or classmates create a communication barrier are encouraged to contact Disability Resources (www.disability.wustl.edu) or talk to their instructor for assistance in determining reasonable adjustments. Adjustments may involve amplification devices, captioning, or clear masks but will not allow for the disregard of mask policies.

• Maintaining physical distancing as needed. While distancing requirements have been removed for vaccinated students, those who are not fully vaccinated are strongly encouraged, for their own health, to maintain a distance of 6 ft from others in the classroom. If you are not able to be vaccinated or have conditions that may put you at increased risk of failed immunity and classroom activities would bring you in frequent proximity to other students, contact your instructor to discuss alternatives.

• Practicing healthy personal hygiene, including frequent handwashing with soap and warm water for at least 20 seconds and/or using hand sanitizer with at least 60% alcohol.

University-Wide Policies:

Reporting Sexual Harassment If a student discusses or discloses an instance of sexual assault, sex discrimination, sexual harassment, dating violence, domestic violence or stalking, or if a faculty member otherwise observes or becomes aware of such an allegation, the faculty member will keep the information as private as possible, but as a faculty member of Washington University, they are required to immediately report it to the Department Chair or Dean or directly to Ms. Cynthia Copeland, the University’s Associate Title IX Coordinator, at (314) 935-3411, cmcopeland@wustl.edu. Additionally, you can report incidents or complaints to the Office of Student Conduct and Community Standards or by contacting WUPD at (314) 935-5555 or your local law enforcement agency. See: Title IX

Reasonable Accommodations for Disabled Students. Washington University in St. Louis supports the rights of enrolled students to a full and equal educational opportunity and, in compliance with federal, state, and local requirements, is committed to reasonable accommodations for individuals with documented disabilities. Disabled students for whom accommodations may be necessary must be registered with, and provide their instructors official notification through, WUSTL’s Disability Resources (www.disability.wustl.edu). Once established, responsibility for disability-related accommodations and access is shared by DR, faculty, and the student. Please contact Disability Resources at 314.935.5970 or disabilityresources@wustl.edu.

Statement on Military Service Leave Washington University recognizes that students serving in the U.S. Armed Forces and their family members may encounter situations where military service forces them to withdraw from a course of study, sometimes with little notice. Students may contact the Office of Military and Veteran Services at (314) 935-2609 or veterans@wustl.edu and their academic dean for guidance and assistance. See: https://veterans.wustl.edu/policies/policy-for-military-students/.

Preferred Name and Gender Inclusive Pronouns In order to affirm each person’s gender identity and lived experiences, it is important that we ask and check in with others about pronouns. This simple effort can make a profound difference in a person’s experience of safety, respect, and support. See: https://students.wustl.edu/genderpronouns-information/, https://registrar.wustl.edu/student-records/ssn-namechanges/preferred-name/.

Emergency Preparedness Before an emergency, familiarize yourself with the building(s) that you frequent. Know the layout, including exit locations, stairwells and the Emergency Assembly Point (EAP). Review the “Quick Guide for Emergencies” that is found near the door in many classrooms for specific emergency information and instructions. For additional Information and EAP maps, visit emergency.wustl.edu. To ensure that you receive emergency notifications, make sure your information and cell phone number is updated in SIS, and/or download. The WUSTL app and enable notifications.

To report an emergency: Danforth Campus: (314) 935-5555 School of Medicine Campus: (314) 362-4357 North/West/South and Off Campus: 911 then (314) 935-5555

Academic Integrity Effective learning, teaching and research all depend upon the ability of members of the academic community to trust one another and to trust the integrity of work that is submitted for academic credit or conducted in the wider arena of scholarly research. Such an atmosphere of mutual trust fosters the free exchange of ideas and enables all members of the community to achieve their highest potential.

In all academic work, the ideas and contributions of others must be appropriately acknowledged and work that is presented as original must be, in fact, original. Faculty, students and administrative staff all share the responsibility of ensuring the honesty and fairness of the intellectual environment at Washington University in St. Louis.

For additional details on the university-wide Undergraduate Academic Integrity policy, please see: https://wustl.edu/about/compliance-policies/academic-policies/undergraduate-studentacademic-integrity-policy/

Resources for Students:

Disability Resources At Washington University we strive to make the academic experience accessible and inclusive. If you anticipate or experience barriers based on disability, please contact Disability Resources at 314.935.5970, disabilityresources@wustl.edu, or visit our website for information about requesting academic accommodations. See: https://students.wustl.edu/disability-resources/. Confidential Resources for Instances of Sexual Assault, Sex Discrimination, Sexual Harassment, Dating Violence, Domestic Violence, or Stalking The University is committed to offering reasonable academic accommodations (e.g. a no-contact order, course changes) to students who are victims of relationship or sexual violence, regardless of whether they seek criminal or disciplinary action. If a student needs to explore options for medical care, protections, or reporting, or would like to receive individual counseling services, there are free, confidential support resources and professional counseling services available through the Relationship and Sexual Violence Prevention (RSVP) Center. If you need to request such accommodations, please contact RSVP to schedule an appointment with a confidential and licensed counselor. Although information shared with counselors is confidential, requests for accommodations will be coordinated with the appropriate University administrators and faculty. The RSVP Center is located in Seigle Hall, Suite 435, and can be reached at resvpcenter@wustl.edu or (314) 935-3445. For after-hours emergency response services, call (314) 935-6666 or (314) 935-5555 and ask to speak with an RSVP Counselor on call. See: RSVP Center.

Bias Reporting and Support System (BRSS) The University has a process through which students, faculty, staff, and community members who have experienced or witnessed incidents of bias, prejudice, or discrimination against a student can report their experiences to the University’s Bias Report and Support System (BRSS) team. See: brss.wustl.edu.

Mental Health Services Mental Health Services’ professional staff members work with students to resolve personal and interpersonal difficulties, many of which can affect a student’s academic experience. These include conflicts with or worry about friends or family, concerns about eating or drinking patterns, and feelings of anxiety, depression, and thoughts of suicide. See: https://students.wustl.edu/mentalhealth-services/. Additionally, see the mental health services offered through the RSVP Center listed above.

WashU Cares WashU Cares specializes is connecting students to mental, medical, financial and academic resources by using supportive case management. We seek to empower students to be successful through life’s challenges and to have ownership of their own experiences. Our services are designed to support Danforth Campus students. If you feel concerned about a student who may need help connecting to resources we accept referrals from all students, faculty, and staff. If you are concerned about a student, you can file a report here: https://washucares.wustl.edu/ and a WashU Cares Case Manager will reach out to you to get more information about your concern.

The Writing Center The Writing Center, located in Olin Library, offers free one-on-one writing tutorials to WashU students, as well as workshops designed to help students become better writers. The Writing Center staff can assist by providing feedback on the strength of an argument, clarity, and organization. Contact them at 935-4981 or writing@wustl.edu. Visit them at: https://writingcenter.wustl.edu/.

The Learning Center The Learning Center works collaboratively with University partners to provide undergraduate students key resources, like academic peer mentoring, to enhance their academic progress. Contact them at https://ctl.wustl.edu/learningcenter/ to find out what support they may offer for your classes.

Center for Diversity and Inclusion (CDI) The Center for Diversity and Inclusion (CDI) supports and advocates for undergraduate, graduate, and professional school students from underrepresented and/or marginalized populations, collaborates with campus and community partners, and promotes dialogue and social change to cultivate and foster a supportive campus climate for students of all backgrounds, cultures, and identities. See: https://diversityinclusion.wustl.edu/.

Gephardt Institute State and local elections can have direct and immediate impacts on our communities and will be happening here in St. Louis and around the country this year. Make sure you stay current on upcoming elections at all levels by registering as a voter, requesting an absentee ballot, or getting election day reminders. You can do all this at http://wustl.turbovote.org for any of the 50 states and Washington D.C. If you are ineligible to vote, you can still participate by referring your friends who are eligible to the TurboVote link to register and by engaging your peers in local issues. If you have any questions about the voting process in the United States, please visit http://washuvotes.wustl.edu.