Lecture 1: 3 January

Introduction:

Course outline, brief history, introduction to dynamical systems

Reading: Strogatz Ch.1

Lecture 2: 6 January

Introduction:

Linear and nonlinear systems; autonomous systems; review of 1st-order ODEs and solution of linear systems

Reading: Strogatz Ch.1

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

Reading: Strogatz Ch.2 (2.4, 2.5)

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
Problemslisted for lectures 3 and 4.

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

Reading: Strogatz Ch.2 (2.8)

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

Somesample 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

Reading: Strogatz Ch.3 (3.1)

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
Problemslisted 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.

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

Reading: Strogatz Ch.3 (3.4)

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 (
extensionto 3 February): theProblemslisted 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.

Lecture 11: 27 January

Pitchfork Bifurcation, Scaling and Nondimensionalization

Overdamped bead on a rotating wire, dimensional analysis, singular perturbations

Reading: Strogatz Ch.3 (3.5)

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

Reading: Strogatz Ch.3 (3.6, 3.7)

Problems:3.6.2, 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

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
Problemslisted 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)

Problems:10.1.11

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)

Problems:5.1.10(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"...]

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:

- logist1.m Iterate logistic map, plot iterates vs n
- logistic_cobweb.m Cobweb construction for logistic map
- logistic_cobweb2.m Cobwebs for logistic map, without transients
- logistic_orbit.m Orbit diagram for logistic map
- logistic_lyapunov.m Lyapunov exponents for logistic map
- Code to iterate general functions f:

Sample function m-files: f.m (logistic map), sinfun.m (sine map)

with corresponding derivatives df.m, dsinfun.m

Composition of functions (2nd, 3rd, 4th iterate maps): f2.m, f3.m, f4.m

Iterate functions without parameters: iter1.m, iter1x.m (iterates vs n),

iter2.m, iter2x.m (cobweb diagrams)

Iterate functions with parameters: iterp1.m, iterp2.m, iterp3.m

Orbit diagram: iter_orbit.m

Lyapunov exponents: iter_lyapunov.m

Lecture 18: 12 February

Logistic Map: Period-doubling route to Chaos

Orbit diagram, computation of 2-cycle, flip bifurcation

Reading: Strogatz Ch.10 (10.2, 10.3)

Problems:10.3.2, 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

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

Reading: Strogatz Ch.10 (10.6)

Problems: Have a good break!

See lecture 17 : Matlab code for iterative maps

- Mid-Semester Break: 21 February

- Homework due 28 February: the
Problemslisted for lectures 15, 16, 17, 18 and 20

Lecture 22: 24 February

Classification of Linear Systems

Eigenvalue/eigenvector method, types of fixed points

Reading: Strogatz Ch.5 (5.2, 5.3)

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

Reading: Strogatz Ch.6 (6.1, 6.2)

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

Reading: Strogatz Ch.6 (6.3)

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
Problemslisted for lectures 22, 23 and 24. Also, think about your project (proposals due March 14).

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"

Reading: Strogatz Ch.6 (6.3, 6.4)

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

Reading: Strogatz Ch.6 (6.5, 6.6)

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

Reading: Strogatz Ch.6 (6.7, 6.8)

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
Problemslisted for lectures 25, 26 and 27; also, project proposals are due.

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

Reading: Strogatz Ch.7 (7.2)

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

Reading: Strogatz Ch.7 (7.3, 7.4)

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

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
Problemslisted 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

Reading: Strogatz Ch.8 (8.0, 8.1)

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

Reading: Strogatz Ch.8 (8.1, 8.2)

Problems:8.1.13, 8.2.1 (exercises)

Exercises: 8.1.12

Lecture 34: 24 March

Hopf Bifurcation

Supercritical and subcritical Hopf bifurcations

Reading: Strogatz Ch.8 (8.2)

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

Lecture 37: 31 March

Global Bifurcations of Cycles, Driven Pendulum

Saddle-node, infinite-period and homoclinic bifurcations of limit cycles; the driven pendulum

Reading: Strogatz Ch.8 (8.4, 8.5)

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:

- lorenzode.m Definition of Lorenz ODEs
- plot_lorenz.m Integration of Lorenz equations, and 2-d and 3-d plots
- plot_lorenz2.m Solution of Lorenz equations for two initial conditions

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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