Spring 2021 - MATH 467 D100

Dynamical Systems (3)

Class Number: 3550

Delivery Method: Remote

Overview

  • Course Times + Location:

    Mo, We, Fr 12:30 PM – 1:20 PM
    REMOTE LEARNING, Burnaby

  • Exam Times + Location:

    Apr 25, 2021
    12:00 PM – 3:00 PM
    REMOTE LEARNING, Burnaby

  • Prerequisites:

    MATH 310.

Description

CALENDAR DESCRIPTION:

Stability and bifurcation in continuous and discrete dynamical systems, with applications. The study of the local and global behaviour of linear and nonlinear systems, including equilibria and periodic orbits, phase plane analysis, conservative systems, limit cycles, the Poincare-Bendixson theorem, Hopf bifurcation and an introduction to chaos. Quantitative.

COURSE DETAILS:



This course will be delivered online. 
You are expected to have access to a reliable internet connection. You will need a computer from which you can download course materials and activities and watch live and/or recorded lectures and participate in live tutorials or workshops.

You will need a camera to take photographs of your work. A phone is acceptable.

Course Outline

Nonlinear Dynamics and Bifurcation

This course is an introduction to the study of dynamical systems. Nonlinear differential equations and iterative maps arise in the mathematical description of numerous systems throughout science and engineering, for instance in physics, chemistry, biology, economics, and elsewhere. Such systems may display complicated and rich dynamical behaviour, and we will develop some linear and nonlinear mathematical tools for their analysis, and consider models in such fields as population biology, ecology, and mechanical and electrical oscillations. Our emphasis throughout will be on the qualitative behaviour of the models, in particular, on the prediction of qualitative change in the nature of the dynamics as a system parameter varies (bifurcation). In this course we will proceed from simpler to more complicated (and more interesting!) systems. We begin with one-dimensional flows, their steady states, stability and bifurcations, and then observe the far more complicated dynamics, including chaos, that may occur in one-dimensional maps. Phaseplane analysis in two dimensions reveals the possibility of oscillations and limit cycles, and we study their bifurcations. As time permits, we will also investigate higher-dimensional dynamical systems, deterministic chaos and strange attractors.

Outline

Introduction to Bifurcation
Fixed points and stability. Saddle node, trans-critical, super-critical and sub-critical pitchfork bifurcation. Linear-systems, Jordan canonical forms, phase plane analysis.

Limit Cycles and Oscillations
Nonlinear centers, index theory, limit cycles, periodic orbits, Poincaré-Bendixon, global attractors, Floquet multipliers.

Normal Forms and Global Bifurcation
Sub- and super-critical Hopf and homoclinic bifurcations, Normal form, center manifold reduction, Poincaré Maps, Normal hyperbolicity.

Chaos and Maps
Maps, chaos, chaotic maps, Lyapunov exponents, area preserving maps renormalization methods.

COURSE DELIVERY

  •  Lecture: synchronous- lectures will be held at fixed times, on-line
  •  Midterm(s): synchronous; date: TBA
  •  Final exam: synchronous; date: TBA


Note: this is a cross-listed course with MATH 767.

Grading

  • Homework (10 sets) 25%
  • Midterm 1 20%
  • Midterm 2 20%
  • Final Exam 35%

NOTES:

THE INSTRUCTOR RESERVES THE RIGHT TO CHANGE ANY OF THE ABOVE INFORMATION.

Students should be aware that they have certain rights to confidentiality concerning the return of course papers and the posting of marks.
Please pay careful attention to the options discussed in class at the beginning of the semester.

REQUIREMENTS:

  • Access to strong and reliable internet.
  • Ability to scan documents (phone app acceptable)
  • Access to webcam and microphone (embedded in computer sufficient)

Materials

REQUIRED READING:

Nonlinear Dynamics & Chaos: With Applications to Physics, Biology, Chemistry, and Engineering
Steven H. Strogatz
Westview Press, 2015
2nd Edition
ISBN: 9780813349107

Registrar Notes:

ACADEMIC INTEGRITY: YOUR WORK, YOUR SUCCESS

SFU’s Academic Integrity web site http://www.sfu.ca/students/academicintegrity.html is filled with information on what is meant by academic dishonesty, where you can find resources to help with your studies and the consequences of cheating.  Check out the site for more information and videos that help explain the issues in plain English.

Each student is responsible for his or her conduct as it affects the University community.  Academic dishonesty, in whatever form, is ultimately destructive of the values of the University. Furthermore, it is unfair and discouraging to the majority of students who pursue their studies honestly. Scholarly integrity is required of all members of the University. http://www.sfu.ca/policies/gazette/student/s10-01.html

TEACHING AT SFU IN SPRING 2021

Teaching at SFU in spring 2021 will be conducted primarily through remote methods. There will be in-person course components in a few exceptional cases where this is fundamental to the educational goals of the course. Such course components will be clearly identified at registration, as will course components that will be “live” (synchronous) vs. at your own pace (asynchronous). Enrollment acknowledges that remote study may entail different modes of learning, interaction with your instructor, and ways of getting feedback on your work than may be the case for in-person classes. To ensure you can access all course materials, we recommend you have access to a computer with a microphone and camera, and the internet. In some cases your instructor may use Zoom or other means requiring a camera and microphone to invigilate exams. If proctoring software will be used, this will be confirmed in the first week of class.

Students with hidden or visible disabilities who believe they may need class or exam accommodations, including in the current context of remote learning, are encouraged to register with the SFU Centre for Accessible Learning (caladmin@sfu.ca or 778-782-3112).