UBC Mathematics 345: Nonlinear Dynamics and Chaos
Section 201; Room: LSK 460; Time: TTh 2:00-3:30
Website for this course:
logbif_vs_r.ode Ode file for Logistice bifurcation diagram.
logbif_vs_r_BlowUP1.ode Ode file for Logistice bifurcation diagram blow-up 1.
logbif_vs_r_BlowUP2.ode Ode file for Logistice bifurcation diagram blow-up 2.
Some Fun Links:
Finishing the homework assignments timely is extremely important for
due in class on date as indicated on each assignment.
- Homework is an essential component of the course (25% of the course grade)
and requires the use of computers
(specific details on how to use the software package XPPAUT
will be given in the Homework Assignments, or see the
XPPAUT main page, but not all features will be used in Math 345).
- Link to the
XPPAUT main page: XPPAUT is free and can be installed on Windows, Linux,
or Mac OS X.
Homework marking rules:
The number of each homework problem should be clearly printed.
Illegible handwritting will result in the loss of marks.
You receive two marks for each collected homework:
one for the completeness and one for the correctness.
Only some important steps and the final answer will be checked.
Rules for Exams and Final Grade Evaluation:
- Homework: 25%
- One Midterm (70 min): 25%.
- Final Exam (150 min): 50%.
- One A4-sized, double sided nd is formula sheet is allowed.
- A small-creen, simple scientific calculator is allowed.
- Late homework assignments receive a grade of 0.
- Missing a quiz/exam results in a mark of 0.
Instructor must be notified within 48 hrs of missed test
to claim medical emergency.
- You must pass the final exam to pass the course!
Text Book and Course Outline (Subject to changes as the course progresses!):
- 18 Jan 2018 (Thursday)
-- First homework due in class.
- 8 Mar 2018 (Thursday)
-- Midterm at class time and location.
- 5 Apr 2018 (Thursday)
-- Last homework due in class.
-- Final exam (Date and location: TBA)
Recommended textbook (required): S. Strogatz, Nonlinear Dynamics and Chaos 2nd Ed. (2015)
Corresponding Textbook Sections
|I. One-Dimensional Flows:
2.0-2.5, 2.8, 3.0-3.5, 4.0-4.4.
- A geometric way of thinking (the phase line)
- Fixed points and stability
- Population growth
- Linear stability analysis
- Existence and uniqueness
- Saddle-node bifurcation, normal form
- Transcritical bifurcation, normal form
- Chemical kinetics
- Pitchfork bifurcation, normal form
- Overdamped bead on a rotating hoop
- Dimensional analysis and scaling
- Overdamped pendulum (with steady applied torque)
|II. Two-Dimensional Flows:
5.0-5.2, 6.0-6.5, 6.7, 7.0-7.3, 8.0-8.3, 8.7
- Linear systems
- Classification of linear systems
- Stability language
- The phase plane
- Existence and uniqueness
- Fixed points and linearization
- Two competing species ("rabbits versus sheep")
- The effect of small nonlinear terms
- Conservative systems
- Pendulum: undamped and damped
- Limit cycles
- Ruling out closed orbits
- Poincare-Bendixson Theorem
- Bifurcations in two dimensions: saddle-node, transcritical and pitchfork bifurcations
- Hopf bifurcations
- Oscillating chemical reactions
- Poincare maps
- One-dimensional maps
- Fixed points and linear stability analysis
- The logistic map
- Invariant sets, attractors, chaos
- Liapunov exponents
- Ruelle plots
- Lorenz equations
- Doubling map