Math 134 – Systems of Differential Equations – Winter 2020

Midterm: Friday Feb 7th, in class.

Final: Monday, March 16th, 11:30 AM-2:30PM

Lectures: MWF 3pm-3:50 pm in MS 5127.

Office Hours: Wed 10-11 am and Thursday 10–11am in 5626 Math Sciences Building.

Overview of the course: In 33B you should have learned to study and solve the behavior of a single system of
differential equations. Here, the focus will be on how the dynamics of the system could be understood qualitatively, how
it can be represented and understood. Instead of asking for exact solutions, we will focus on understanding things like “How does the
system behave for large time given an initial condition?” One might be motivated by asking the old question: “Is the solar system stable,
or will the planets eventually diverge out of the sun?”. This requires understanding stability, equilibria and phase diagrams.

We will see what types of phase diagrams can appear in a system, and how they can be studied based on the system. Often, a systems
behavior will depend on a parameter, and we will study “bifurcation theory”, which studies how the phase diagram changes
with the parameter. Interesting phenomena occur, and this will depend on the dimensionality of the system. Higher dimensional
systems, unsurprisingly, exhibiting more complexity.

In the beginning we will work to understand one-dimensional systems, and through much work, at the end of the course
we understand two-dimensional systems. The complexity of this general field is exhibited by this limitation; in ten weeks
we only reach two dimensional systems. For these two dimensions, special geometric and topological tools
will enable us to give interesting conclusions.

Course Book: S. Strogatz, Nonlinear Dynamics and Chaos (2nd Ed.), Perseus Books Group. Sections below refer to this book.

Supplementary reading: J. Crawford, Introduction to Bifurcation Theory, Reviews of Modern Physics, vol. 6

Skills to learn and that will be tested:

  • Sketching phase diagrams and understanding their dynamics.
  • Discuss behavior of non-linear dynamical systems qualitatively. Describe possible behaviors.
  • Give examples of systems with given behaviors.
  • Undestand implications of existence and uniqueness for phase diagrams.
  • Recognize and analyze stability of critical points, and sketch phase diagrams in their vicinity.
  • Understand and describe two-dimensional dynamics, including limit cycles and indices.
  • Appreciate how topology restricts the dynamics of the system.

Instructor: Sylvester Eriksson-Bique, 5262 Math Sciences Building.


  • Stanley Palasek: 3A Thursday 3pm-3:50pm MS 5127

Exams: One in-class midterm: Friday Feb 7th, in class. Three-hour final: Monday, March 16th, 11:30 AM-2:30PM

Bring student ID to both midterms and the final. There will be no make-up exams.
No calculators, notes, or books will be permitted in any exam.
You may bring food/snacks for the Final since it is during lunch, but please be respectful of fellow exam takers.

Homework: There will be weekly homework. It is due in the beginning of the class on Friday. Further information is given below.

  • No late homework will be accepted.
  • Graded for a total of 20 points each set, 5 for each graded problem (usually three) and 5 for completeness of remainder.
  • Write your name, ID number.
  • Staple your pages! May submit online as pdf on CCLE, but please remember to SUBMIT.
  • The weakest homework score will be omitted.

Discussion sections: Attendance is strongly encouraged. You may discuss and work on similar problems as HW.


  • Normal: Homework, 25%; Midterm 35%; Final 40%.
  • Given a compelling (and documented) reason to miss the midterm: Homework, 30%; Final 70%.
  • Final exam can not be missed. If missed, it will lead to an incomplete or F, and require retaking class/final in a different quarter.

Extra credit: This class will involve many complex concepts. I highly recommend you engage in active discussion on the class
piazza site. You may receive 1-2 % extra credit for this. Also, you can during the last three weeks of class bring show to me one
of the following for 1-2 % extra credit. Evaluations are final, and will not be increased. I may give a 0% and ask improvements if the work is not sufficient
for extra credit, a second sincere attempt would lead to at least 1 % more credit.

  • Given a system of your choice with at least one parameter, describe its phase diagram and bifurcations you witness. Use at least
    5 of the terms in this class. Explain how the phase diagram comes from the equation.
  • Take at least a two dimensional system, and draw its dynamics using your favorite programming language (python, matlab recommended) and describe it to me.
  • Take a theorem from the class material (e.g. Poincare-Bendixson theorem) and present its proof. I can help with finding references.
  • Give a short ~5-10 min quick talk on how a dynamical system similar to the ones in class is used in an application area of your choice. Mention
    at least two concepts/terms from this class that are relevant in this application and explain what they mean in this context.
  • Create your own challenge. You can do something similar. It must pertain to this class, must demonstrate your understanding of
    some key concepts in this class, must receive approval from me (say after class), and must
    be presentable in about 10 minutes. Email submissions can be considered as long as you discuss ahead of time.

Inclusivity: This course is welcoming to everyone and from any background. If you at any time feel
a policy of this course causes excessive hardship or is difficult to satisfy for whatever reason (e.g. office hours
conflict with work, or exams conflict with religious holiday), please do let me know ahead of time and we will
work to find an accommodation. If you encounter problems in your personal life, or medical issues, please do reach out for help,
and I can also do my best to forward you to university resources that can help you.

CAE brochure

My goal is to foster an inclusive, supportive and encouraging enviroment for you to learn. Many of us suffer from various degrees
of learning difficulties, exam stress etc. Please be aware that the university has many resources to help with challenges you might face
and that I as an instructor will do my best to assist and accomodate different learners. Please do not be afraid to reach out to a councelor,
me or your TA for contact information to various services. Also, please review the attached brochure.


Tentative course schedule

Tentative course schedule

Week Date Section Topic
1 Mon Jan 6th-10th 1.0-2.3 Overview, definition of dynamical system, importance
2 Jan 13th-17th 2.4-3.1 One dimensional flows, potentials, numerics, intro to bifurcations
3 Jan 21st-24th 3.0-3.5 MLK Day on Monday Transcritical and pitchfork bifurcations.
4 Jan 27-Jan 31st 3.5-4.3 Dimensional analysis, Imperfect bifurcations, dynamics on the circle.
5 Feb 3rd-Feb 5th 4.4-5.1 Oscillator examples and intro to two-dimensional lin systems
MIDTERM Midterm Friday Feb 7th 1–4.6 Up to Section 4.6.
6 Feb 10th-Feb 14th 5.2-6.3 Classification of linear systems, intro to two-dim non-linear systems and equilibria and stability
7 Feb 18th-Feb 21st 6.5-6.7 Monday Presidents Day Holiday Equilibria,stability, special systems, heteroclinic, homoclinic orbits, applications
8 Feb 24th-Feb 28th 6.8-7.2 Index theory and winding numbers, limit cycles and ruling out limit cycles. Liapunov functions.
9 Mar 2nd-Mar 6 th 7.3-8.3 Existence of closed orbits, Poincare-Bendixson theorem, Bifurcation in two (and higher) dimensions, Hopf bifurcation
10 Mar 9th-Mar 13th 8.4 Global bifurcation of cycles and review
FINAL Monday March 16th Cumulative 11am-2:30pm Location TBD

Additional policies

Additional Links

Notice about academic integrity

From the office of the Dean of Students:

“With its status as a world-class research institution, it is critical that the University uphold the highest
standards of integrity both inside and outside the classroom. As a student and member of the UCLA
community, you are expected to demonstrate integrity in all of your academic endeavors. Accordingly,
when accusations of academic dishonesty occur, The Office of the Dean of Students is charged with
investigating and adjudicating suspected violations. Academic dishonesty includes, but is not limited
to, cheating, fabrication, plagiarism, multiple submissions or facilitating academic misconduct.”

Students are expected to be aware of the University policy on academic integrity in the UCLA Student
Conduct Code:

Please note the sections on (1) cheating, (2) plagiarism, and (3) unauthorized study aids.
Violation of course policy involving plagiarism, cheating, or possession of course materials during
exams will be referred to the Dean of Students, who will be encouraged to take strong action. Do not
cheat! The penalties can be very harsh. Do not believe it if you hear that “everyone does it.” You
generally do not hear about the punishments because they are kept confidential. If you are found
responsible by the Dean of Students for violating course policy, cheating on any course materials, or
giving or receiving unauthorized help, a zero will be assigned for the entire assignment. No exceptions
will be made! Past examples of penalties also include loss of an entire term of credit and suspension for
several terms. If you plan to apply to graduate or professional school, such a negative mark on your
record may be a major obstacle to admission.

No cell phones are allowed during exams. They must be left in your bag and turned off, or submitted to
the designated TA/proctor. Students may not use a cell phone as a clock to keep time, nor as a
calculator. No hats are allowed in the testing room; they must be left in your bag.

Notice about sexual harassment, discrimination, and assault

Title IX prohibits gender discrimination, including sexual harassment, domestic and dating violence,
sexual assault, and stalking. Students who have experienced sexual harassment or sexual violence can
receive confidential support and advocacy from a CARE advocate:

The CARE Advocacy Office for Sexual and Gender-Based Violence

1st Floor, Wooden Center West

(310) 206-2465

You can also report sexual violence or sexual harassment directly to the University’s Title IX

Kathleen Salvaty

2241 Murphy Hall

(310) 206-3417

%d bloggers like this: