Assignment schedule:

THESE WEBPAGES DESCRIBE HOW THIS COURSE WILL BE RUN.
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Homework: Typically there will be one homework per week. The HW will be posted here every Thursday (at the latest) and is due the following Thursday and must be handed in as soon as you enter the classroom before the lecture starts . In some occasions the HW will be posted a few days before Thursday so you are encouraged to start on it earlier. Late HW will not be accepted.

Take-home exams: There will be two take home exams during the semester. They will replace the homework for that week. Take-home exams will be posted and due the same way the homework is. Late take-home exams will not be accepted.

 

Assignments: [bottom]

Week#
 Topics: Sec.: Exercises: Due:
01
22/01
Chap 1: Overview:
Introduction
1.0-1.3
    HW#00:
  • Go over Matlab [PDF] tutorial [For students without previous knowledge on Matlab (if you are an "expert" in Matlab just send me an email indicating so)]:
    Type in all instructions in Matlab (follow the whole tutorial) and submit to me by email a diary of it (to do the diary: do "diary FirstName_LastName_M638.txt" as the first instruction and this will save all your inputs in that file)
Th 29 Jan
02
27-29/01
Chap 2: Flows on the line:
Fixed points and stability +
Population growth
Linear Stability Analysis +
Existence and Uniqueness +
Impossibility of Oscillations +
Potentials
2.0-2.3.
2.4-2.8
    HW#01:
  • Ex: 2.2.3 + 2.2.4 + 2.2.8 + 2.2.9
  • Ex: 2.3.3 + 2.3.4
  • Ex: 2.4.4 + 2.4.8
  • Ex: 2.5.1 + 2.5.3
  • Ex: 2.7.1 + 2.7.6
Th 05 Feb
03
03-05/02
Solving Eqns on the Computer +

Chap 3: Bifurcations:
Saddle-node Bifurcations +
Transcritical bifurcations +
Laser Threshold +
Pitchfork bifurcations
3.1-3.4
    HW#02:
  • Ex: 3.1.1 + 3.1.4
  • Ex: 3.2.1 + 3.2.6 + 3.2.7
  • Ex: 3.3.1
  • Ex: 3.4.2 + 3.4.4
  • For dx/dt = r x + x3 - x5 find the location of the saddle nodes bifurcations and do a Taylor approximation about this bifurcation point (and rescaling) to obtain the familiar normal form for the saddle node bifurcation.
Th 12 Feb
04
10-12/02
Imperfect bifurcations +
Insect outbreaks

Chap 5: Linear Systems:
Definitions and Examples
3.4, 3.6-3.7
5.1-5.2
    HW#03:
  • Ex: 3.6.2 + 3.6.4
  • Ex: 3.7.3 + 3.7.4 + 3.7.6
Th 19 Feb
05
17-19/02
Classification of linear systems +
Stability
5.1-5.2
    HW#04:
  • Ex: 5.2.2 + 5.2.12 + 5.2.13
  • Bonus1 = 5.2.14
  • Ex: 5.3.2 + 5.3.4 + 5.3.6
Th 26 Feb
06
24-26/02
Chap 6: Phase Plane:
Phase Portraits +
Existence and Uniqueness
Fixed Points and Linearization +
Competition Models +
Conservative Systems
6.1-6.5
    HW#05:
  • Ex: 6.1.8 + 6.1.14
  • Ex: 6.2.1 + 6.2.2 (also plot orbits for b) and c) using computer)
  • Ex: 6.3.9 + 6.3.16
  • Ex: 6.4.6
Th 05 Mar
07
03-05/03
Reversible Systems +
Pendulum +
Index Theory

Chap 7: Limit Cycles:
Ruling out closed orbits +
Gradient systems
Lyapunov functions +
6.6-6.8
7.1-7.2
    HW#06:
  • Ex: 6.5.9 (read the paragraph on Hamiltonians on p187) + 6.5.10 + 6.5.14
  • Ex: 6.6.8 + 6.6.9 (k=1,2 and N=2)
  • Ex: 6.8.7 + 6.8.12
Th 12 Mar
08
10-12/03
Dulac's criterion +
Poincaré-Bendixson theorem +
Liénard systems
7.2-7.4
    HW#07:
  • Ex: 7.2.6 + 7.2.8 + 7.2.12
  • Ex: 7.3.1 + 7.3.2 + 7.3.4
  • Ex: 7.4.2
Th 19 Mar
09
17-19/03
Relaxation oscillations +
Weakly nonlinear oscillators +
Regular perturbation theory +
Two-timing +
7.5-7.6
    HW#08:
  • Ex: 7.5.6 + 7.5.7
  • Ex: 7.6.11 + 7.6.14 + 7.6.15
  • Bonus1 = 7.6.19
Tu 07 Apr
10
24-26/03
Two-timing +
Averaged equations
7.6
     
 
--
31/03-02/04

SPRING BREAK
 
NO CLASSES
 
11
07-09/04
Chap 8: Bifurcations in 2D:
Saddle node
Supercritical Hopf +
Subcritical Hopf +
Oscillating chemical reactions +
Global Bifurcations of Cycles
8.1-8.4
    HW#09:
  • Turn in a paragraph describing in detail your project. Include references if possible.
  • Ex: 8.1.12
  • Ex: 8.2.[2+3] + 8.2.11
  • Ex: 8.3.1
  • Ex: 8.4.3
  • Ex: 8.4.[5,6,7,8,9,10,11]
    • [Note: in 8.4.5 the r'-equation should have a sin(φ) and not a cos(φ) (some versions of the book have this typo)]
Tu 21 Apr
12
14-16/04
Global Bifurcations of Cycles +
Coupled Oscillators and Quasiperiodicity +
Poincaré Sections +

Chap 9: Lorenz Equations:
Chaotic waterwheel +
Lorenz eqns +
Simple properties of Lorenz eqns
8.4, 8.6-8.7
9.0-9.2
    HW#10:
  • Work on projects!!!
  • Ex: 8.6.1 + 8.6.7
  • Ex: 8.7.3
Th 23 Apr
13
21-23/04
More Lorenz eqns:
Nonlinearity + Symmetry, Volume contraction + Fixed points + Linear Stability + Global stability of origin
Chaos on a Strange Attractor +
Lorenz Map +
Exploring Parameter Space +
Computing Lyapunov exponents using continuous QR factorization: [paper , notes]
8.7
9.2-9.5
    HW#11:
  • Work on projects!!!
  • Write a program to compute the Lyapunov exponents for a three-dimensional systems using the QR method described in the paper by Geist that I distributed in class. Submit to me by email a copy of your code ready to run on the Lorenz equations.
  • Use the code to compute the Lyapunov Exponents of the Lorenz equations and check whether their sum is correct. Turn in plots for the time series of the Lyapunov exponents vs. time and a typical orbit of the attractor. Also report the values that the Lyapunov exponents converge to.
  • Do same for the Rossler attractor for (a,b,c)=(0.1,0.1,14)
Th 30 Apr
14
28-30/04
Hamiltonian systems +
The extensible pendulum +
The magnetic pendulum +
Transition to chaos +
Lorenz systems by Prof Bo-Wen Shen
Notes on Extensible Pendulum
  • Work on projects!!!
  
15
04/05
Synchronization +
Project presentations 		Notes on Synchronization
     
 

Th 6 May
Fr 7 May

 Project Presentations:
(1) Thu May 7th : 9:15-11:00 @ GMCS-328
(2) Fri May 8th : 11:00-1:00 @ GMCS-405
 
   Arrive early to test equipment/presentations
Th 6 and Fr 7 of May

Bonus1: these exercises are for extra credit for undergrads and required for grad students.
 Bonus2: these exercises are for true extra credit (idependently if you are undergraduate or graduate student).