Simon Fraser University Department of Mathematics
Spring 2003

## MATH 467-3: Dynamical Systems

### Course Schedule:

Week  0:

• Lecture 1: 3 January
Introduction:
Course outline, brief history, introduction to dynamical systems

Week 1:

• Lecture 2: 6 January
Introduction:
Linear and nonlinear systems; autonomous systems; review of 1st-order ODEs and solution of linear systems

• Lecture 3: 8 January
Flows on the Line - a Geometric View:
Direction field, phase line, fixed points, stability, population growth and logistic equation

Reading: Strogatz Ch.2 (2.0, 2.1, 2.2, 2.3)
Problems: 2.2.7, 2.2.8, 2.2.13, 2.3.2, 2.3.3
Exercises (optional): 2.2.1, 2.2.6, 2.2.9, 2.2.10, 2.3.1

• Lecture 4: 10 January
Linear Stability:
Linear stability analysis, existence and uniqueness

Problems: 2.4.7, 2.4.9, 2.5.1, 2.5.2
Exercises: 2.4.1, 2.4.8, 2.5.3, 2.5.4

• Homework due 17 January: the Problems listed for lectures 3 and 4.

Week 2:

• Lecture 5: 13 January
Potentials; Introduction to Bifurcation:
Nonexistence of oscillations, potential; introduction to bifurcation, saddle-node bifurcation

Reading: Strogatz Ch.2, 3 (2.6, 2.7, 3.0, 3.1)
Problems:  2.7.6, 3.1.3
Exercises:  2.6.1, 2.7.3, 2.7.7, 3.1.1

• Lecture 6: 15 January
Numerical Integration of ODEs:
Principles of one-step methods, Euler's method, error analysis and stability, improved numerical methods

Problems:  2.8.2(c), 2.8.3, 2.8.4, 2.8.5
Exercises:  2.8.6, 2.8.8
Resources:
Introduction to Maple (Maple worksheet maple_intro.mws)
Maple: Waterloo Maple Software web site
Introduction to Matlab (PDF document, 40 pages, by Kermit Sigmon, U. of Florida)
Matlab: Mathworks web site
dfield and pplane: ODE software for Matlab (John Polking, Rice University)
dfield and pplane Java applets
Some sample code:
Sample Euler method implementation (Maple worksheet euler_sample.mws;
a simpler approach euler_sample2.mws)
Notes on a Matlab implementation of the Euler method (PDF)

• Lecture 7: 17 January
Bifurcations of Fixed Points: Saddle-Node Bifurcation, Normal Forms

Saddle-node bifurcation, local analysis, normal forms, near-identity transformation

Problems:  3.2.4, 3.2.6
Exercises:  3.1.5, 3.2.3, 3.2.7 (read this result)

• Homework due 24 January: the Problems listed for lectures 5, 6, and 7.
For 3.1.3 and 3.2.4, also use Maple or Matlab to plot some representative numerical solutions of x(t) for three values of r, below, at and above the bifurcation value.
Some sample implementations of the Euler method for Maple and Matlab are given for lecture 6.
For 2.8.3(c), you should plot ln E vs. ln
Dt.

• Solutions to Homework 1: Maple worksheet sol1.mws, PDF, HTML.

Week 3:

• Lecture 8: 20 January
Bifurcations of Fixed Points: Transcritical Bifurcation
Curves of zeros and implicit function theorem; transcritical bifurcation, laser threshold

Reading: Strogatz Ch.3 (3.1, 3.2, 3.3)
Problems:  3.2.5, 3.3.1
Exercises:  3.2.3; read the result of 3.2.7 (try it if you like!)

• Lecture 9: 22 January
Bifurcations with Symmetry: Pitchfork Bifurcation
Supercritical and subcritical pitchfork bifurcation, phase transitions, hysteresis

Problems:  3.4.8, 3.4.12, 3.4.14
Exercises:  3.4.2, 3.4.13, 3.4.16; 3.4.11 is interesting

• Lecture 10: 24 January
Software for Exploration of Dynamical Systems
Maple and Matlab routines, including DEtools (Maple), dfield6 and pplane6 (Matlab), and computation of bifurcation curves
Problems Numerical bifurcation diagrams (PDF file)
You may wish to look at Section 3.6 (dynamical systems depending on two parameters, and stability diagrams) while attempting this problem.
Matlab code:

Resources:

• See lecture 6 for introductions to Maple and Matlab, and links
• dfield and pplane: ODE software for Matlab (John Polking, Rice University)
• Maple in-class demonstration: demo.mws
• Matlab routines to approximate curves of stable equilibria:
superpf.m, mysuperpf.m   (try superpf('mysuperpf',-2,0.8,10,1.e-4))
subpf.m, mysubpf.m         (try subpf('mysubpf',-1,-0.1,3,1.e-4) )
(These routines attempt to find the stable branches of bifurcation diagrams by integrating dx/dt = f(x,r) forward in x, and letting r vary slowly, via dr/dt = epsilon; run the Matlab scripts to estimate bifurcation diagrams, and see how successful (or not!) this approach is, compared with the method explored in the homework.)
• Homework due 31 January (extension to 3 February): the Problems listed for lectures 8, 9 and 10.  In particular, you should work on the computational project (using Matlab), described in this PDF file, since this may take some time to complete.

• (Partial) Solutions to Homework 2: Maple worksheet sol2.mws.

Week 4:

• Lecture 11: 27 January
Pitchfork Bifurcation, Scaling and Nondimensionalization
Overdamped bead on a rotating wire, dimensional analysis, singular perturbations
Problems: 3.5.6, 3.5.7
Exercises: 3.5.1, 3.5.2, 3.5.5, 3.5.8

• Lecture 12: 29 January
Imperfect Bifurcations
Imperfect bifurcations, cusp catastrophe; insect outbreak model

Problems3.6.2, 3.6.3, 3.7.4 (look at 3.7.3), 3.7.6
Exercises: 3.5.4, 3.6.1, 3.7.1, 3.7.3

• Lecture 13: 31 January
Flows on the Circle
Examples, uniform oscillator, nonuniform oscillator

Reading: Strogatz Ch.4 (4.0, 4.1, 4.2, 4.3)
Problems: 4.1.8
Exercises: 4.1.1, 4.1.5, 4.2.1

Week 5:

• Lecture 14: 3 February
Nonuniform Oscillator
Oscillation period, ghosts and bottlenecks, square-root scaling, overdamped pendulum
Reading: Strogatz Ch.3 (4.3, 4.4) - also read 4.5 (and 4.6 if you are interested)
Problems: 4.3.5, 4.5.1
Exercises: 4.3.1, 4.3.2, 4.3.3, 4.3.9, 4.4.1; 4.4.4

• Homework due 10 February: the Problems listed for lectures 11, 12, 13 and 14.
(Remember: there is no homework set next week; the next set will be due 28 February)
For problem 4.5.1, you should read section 4.5.

• Lecture 15: 5 February
One-dimensional Maps
Iterative maps, fixed points and stability, periodic orbits, cobwebs, the logistic map

Reading: Strogatz Ch.10 (10.0, 10.1, 10.2)
Problems10.1.11
Exercises: 10.1.1, 10.1.6, 10.1.10, 10.1.12, 10.2.1, 10.2.2
See lecture 17 : Matlab code for iterative maps

• Lecture 16: 7 February
Flows in the Phase Plane and Linear Systems
Two-dimensional systems, trajectories and the phase plane, linear systems, phase portraits and stability

Reading: Strogatz Ch.5 and 6 (6.1, 5.0, 5.1)
Problems5.1.10(a,c,e), 5.1.11(a,c,e)
Exercises: 5.1.4, 5.1.9, 5.1.13
[Note: compared to the natural progression of the course material and syllabus, this lecture and the next few are "switched"...]

Week 6:

• Lecture 17: 10 February
Logistic Map: Qualitative Behaviour
Fixed points, stability, transcritical bifurcation
Reading: Strogatz Ch.10 (10.1, 10.2, 10.3)
Problems: 10.2.4, 10.2.6
Exercises: 10.2.1, 10.2.5, 10.2.8
Resources - Matlab code:

• Lecture 18: 12 February
Logistic Map: Period-doubling route to Chaos
Orbit diagram, computation of 2-cycle, flip bifurcation

Problems10.3.2, 10.3.4
Exercises: 10.3.6, 10.3.11
See lecture 17 : Matlab code for iterative maps

• Lecture 19: 14 February
Midterm Exam 1
Covering material from Chapters 1-4, and 10.1-3

Week 7:

• Lecture 20: 17 February
Logistic Map and Chaos
Stability of 2-cycle, periodic windows, 3-cycle and intermittency, tangent bifurcation; characterization of chaos, Liapunov exponents
Reading: Strogatz Ch.10 (10.3, 10.4, 10.5, 10.6)
Problems: 10.3.9, 10.3.10, 10.5.3, 10.5.4
Exercises: 10.4.6, 10.6.1, 10.6.3
See lecture 17 : Matlab code for iterative maps

• Lecture 21: 19 February
Universality in Chaos

Universality for unimodal maps, Feigenbaum constant, period-doubling in convection and the Rossler system; review of Midterm 1

ProblemsHave a good break!
See lecture 17 : Matlab code for iterative maps

• Mid-Semester Break: 21 February

• Homework due 28 February: the Problems listed for lectures 15, 16, 17, 18 and 20

Week 8:

• Lecture 22: 24 February
Classification of Linear Systems
Eigenvalue/eigenvector method, types of fixed points
Problems: 5.2.5, 5.2.13 (do 5.2.1, 5.2.2 if you need review on solving linear systems)
Exercises: 5.2.1, 5.2.2, 5.2.3, 5.2.7, 5.2.9

• Lecture 23: 26 February
Phase Portraits
The phase plane, the existence/uniqueness theorem and consequences
Problems: 6.1.5, 6.1.13
Exercises: 6.1.1, 6.1.9, 6.1.12, 6.1.14, 6.2.2

• Lecture 24: 28 February
Linearization
Linearization, relation between linear and nonlinear system, drawing a phase portrait
Problems: 6.3.4, 6.3.10, 6.3.16
Exercises: 6.3.1, 6.3.9, 6.3.12

• Homework due 7 March: the Problems listed for lectures 22, 23 and 24.  Also, think about your project (proposals due March 14).

Week 9:

• Lecture 25: 3 March
Structural Stability and Hyperbolicity, Competition Models
Polar coordinates, example where linearization fails, hyperbolic fixed points, Lotka-Volterra competition model "rabbits vs. sheep"
Problems: 6.1.8, 6.1.10, 6.4.7 (lasers again)
Exercises: 6.4.1, 6.4.4, 6.4.6

• Lecture 26: 5 March
Conservative Systems
Conservative systems, potentials, nonlinear centres, reversible systems
Problems: 6.5.6 (epidemics again), 6.5.13, 6.5.19 (predator-prey model)
Exercises: 6.5.2, 6.5.8, 6.5.15-18 (more on the bead on a rotating hoop), 6.6.1, 6.6.5

• Lecture 27: 7 March
Pendulum, Index Theory
Undamped and damped pendulum, cylindrical phase space, introduction to index theory
Problems: 6.7.2 (pendulum driven by constant torque), 6.8.2, 6.8.3, 6.8.4, 6.8.6
Exercises: 6.8.1; 6.8.10, 6.8.11

• Homework due 14 March: the Problems listed for lectures 25, 26 and 27; also, project proposals are due.

Week 10:

• Lecture 28: 10 March
Index Theory, Limit Cycles
Properties of the index, examples of limit cycles, van der Pol oscillator
Reading: Strogatz Ch.6 and 7 (6.8, 7.0, 7.1)
Problems: 7.1.3 (Try 7.1.9!)
Exercises: 7.1.2, 7.1.8

• Lecture 29: 12 March
Conditions for Nonexistence of Limit Cycles
Gradient systems, Liapunov functions, Bendixson's and Dulac's criterion
Problems: 7.2.7, 7.2.10, 7.2.13
Exercises: 7.2.1, 7.2.5, 7.2.8, 7.2.9, 7.2.11, 7.2.14, 7.2.16

• Lecture 30: 14 March
Poincare-Bendixson Theorem
Existence of closed orbits, trapping regions, glycolytic oscillations, no chaos in the plane
Problems: 7.3.1, 7.3.2, 7.3.4, 7.3.9 (also try 7.3.7)
Exercises: 7.3.6, 7.3.7, 7.3.11

Week 11:

• Lecture 31: 17 March
Lienard Systems, Relaxation Oscillations
The Lienard plane, van der Pol oscillator
Reading: Strogatz Ch.7 (7.3, 7.4, 7.5)
Problems: 7.4.2, 7.5.4
Exercises: 7.5.1, 7.5.7

• Homework due 24 March: the Problems listed for lectures 28, 29, 30 and 31.

• Lecture 32: 19 March
Bifurcations (Revisited)
Bifurcations of fixed points in two dimensions: saddle-node, transcritical and pitchfork
Problems: 8.1.11 (exercise)
Exercises: 8.1.1, 8.1.3, 8.1.5, 8.1.6

• Lecture 33: 21 March
Bifurcations of Fixed Points
Saddle-node, transcritical and pitchfork bifurcations; Hopf bifurcations
Problems: 8.1.13, 8.2.1 (exercises)
Exercises: 8.1.12

Week 12:

• Lecture 34: 24 March
Hopf Bifurcation
Supercritical and subcritical Hopf bifurcations
Problems: 8.2.8 (exercise)
Exercises: 8.2.9, 8.2.12

• Lecture 35: 26 March
Creation of Oscillations
Hopf bifurcations, oscillating chemical reactions, global bifurcations
Reading: Strogatz Ch.8 (8.2, 8.3, 8.4)
Problems: 8.3.1 (HW 9), 8.4.3 (exercise)
Exercises: 8.3.2

• Lecture 36: 28 March
Midterm Exam 2
Covering material from Chapters 5-8

Week 13:

• Lecture 37: 31 March
Global Bifurcations of Cycles, Driven Pendulum
Saddle-node, infinite-period and homoclinic bifurcations of limit cycles; the driven pendulum
Exercises: 8.4.3, 8.4.12

• Lecture 38: 2 April
Lorenz Equations
Hysteresis in the driven pendulum, introduction to the Lorenz equations
Reading: Strogatz Ch.8 and 9 (8.5, 9.1, 9.2)
Exercises: 8.5.2, 9.1.4
Matlab code:
• Lecture 39: 4 April
Quasiperiodicity and Chaos
Flows on the torus, Poincare map, chaos in the Lorenz equations
Reading: Strogatz Ch.8 and 9 (8.6, 8.7, 9.2, 9.3, 9.4)
Problems: 9.2.1, 9.2.2, 9.2.3, 9.2.4, 9.3.2, 9.3.4, 9.3.5 (HW 9)
Exercises: 8.7.1, 8.7.5, 9.2.5, 9.5.1

The problems listed above are the homework problems corresponding to each lecture.  Problems for which I have made a note or given a hint are hyperlinked to the Homework Notes page.  Exercises are optional, but I encourage you to look at them; they are usually designed as a review, a simpler introduction to the basic ideas of a section, an alternative perspective to ideas discussed in class, or additional practice.

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