Spring 2021 - MATH 467 D100
Dynamical Systems (3)
Class Number: 3550
Delivery Method: Remote
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
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.
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.
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.
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.
- 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.
- Homework (10 sets) 25%
- Midterm 1 20%
- Midterm 2 20%
- Final Exam 35%
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.
- Access to strong and reliable internet.
- Ability to scan documents (phone app acceptable)
- Access to webcam and microphone (embedded in computer sufficient)
Nonlinear Dynamics & Chaos: With Applications to Physics, Biology, Chemistry, and Engineering
Steven H. Strogatz
Westview Press, 2015
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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 (email@example.com or 778-782-3112).