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 Wednesday (at the latest) and is due the following Wednesday 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 Wednesday 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.

Review instructions: Some HW will consist in reviewing a research article. Please follow this link for the [Review instructions].

 

Assignments: [bottom]

Week #
and HW#
  Topics:
Sec.:
  Exercises:
Due:
01
29-31/08
Introduction +
1D maps +
Cobweb plots +
1.1-1.2
  • Go over Matlab 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_M342A.txt" as the first instruction and this will save all your inputs in that file)
  • You can download here the first chapter of the book [chap1.pdf]
  • Reproduce cobweb plots:
    Fig 1.1 (x0 = 0.01) +
    Fig 1.2 (x0 = 0.01, 0.8, 1.01, -0.01) +
    Fig 1.3 (x0 = 1.6, 1.8, -0.01, 0.01)
    using Matlab and explain your results
We 07 Sep
02
07/09
Stability of fixed points +
Basins of attraction +
Periodic orbits
1.3, 1.4
  • Ex T1.3, T1.4 (p12-13).
  • Ex 1.1, 1.2, 1.3 (p36).
  • Extra/Grad Credit*: state and proof a nonlinear version of the stability theorem (Theo 1.5) when linear stability fails (i.e., |f '(x*)|=1).
  • (Remember to ellaborate and explain your answers. This is true for ALL assignments)
We 14 Sep
03
12-14/09
Periodic orbits +
Logistic map family +
The logistic map +
Sensitive dependence to ICs
1.4-1.8
  • Reproduce Figs. 1.6 and Fig 1.7 (p19-20 of our textbook).
  • Ex T1.5, T1.7, T1.8, T1.9 (p14-24).
  • Ex 1.4, 1.5 (p36).
  • Review: 'Simple mathematical models with very complicated dynamics', R. May, Nature 261 (1976) p459. [PDF]
    [Review instructions]
  • Ex 1.7, 1.10, 1.12 (in (b) use cobweb) (p36-37).
  • Extra/Grad Credit*: Review: 'Chaos in the cubic mapping', T. Rogers and D.C. Whitley, Mathematical Modelling 4 (1983) pp.9-25. [PDF]
    [Review instructions]
We 21 Sep
04
19-21/09
Itineraries, Symbolic Dynamcs +
2D maps:
Poincaré sections +
Hénon map +
Sink, sources and saddles
2.1-2.2
  • Read/Study eigenvalue notes from "Linear Algebra and Diff. Eqns. Using Matlab" by Golubitsky and Dellnitz.
    [2x2 matrices] [NxN matrices]
  • Reproduce Fig 5 of the article "Chaos in the cubic mapping" (see above, [PDF]).
    Note: the procedure on how to produce bifurcation diagrams is briefly explained in the first paragraph in p18 of our textbook.
We 28 Sep
05
26-28/09
2D maps:
Sink, sources and saddles +
Linear maps +
Coordinate Changes +
2.2-2.4
  • Ex 2.3, 2.5, 2.6
  • Ex T2.2, T2.3, T2.4, T2.5, T2.7
  • Extra/Grad Credit*: Reproduce Fig. 2.3 (p51 of our textbook).
We 05 Oct
06
03-05/10
2D maps:
Jacobian
Periodic orbits +
Bifurcation diagram of Hénon map +
Stable and Unstable manifolds Chaos:
Lyapunov exponents +
chaotic orbits +
Tent map +
2.5-2.6,
3.1-3.2
    No HW this week.
 
07
10-12/10
 MT#1: In class part : Monday Oct 10 +
MT#1: take home part : work on MT (no class on Wednesday Oct 12)
 
  • Take home MT#1:
Mo 17 Oct
Mo 10 Oct
 Midterm #1

All: Chaps 1 + 2

  • Please arrive EARLY (1:50pm).
Mo 10 Oct
08
17-19/10
Conjugacy +
Basins of Attraction +
Fractals +
Cantor sets +
3.3-3.5,
4.1
  • For the logistic map f(x)=ax(1-x), write a program to compute the Lyapunov exponent and reproduce fig. 6.3.b. Also plot the bifurcation diagram for the same a-window and compare/discuss both plots.
  • Do the same for the cubic map of the extra credit of HW#03: f(x) =a x3 + (1-a) x, for a in [2,4].
  • Ex 3.1, 3.3, 3.4, 3.15
We 26 Oct
09
24-26/10
Iterated Function Systems +
Determinstic Fractals +
Fractal basins +
Mandelbrot set +
Julia sets +
Fractal dimension +
Box Counting Dimension +
4.1-4.6
  • Ex T4.8, T4.9, 4.7, 4.14
  • Write a code to compute the box counting dimension and compute it for the Hénon map. (Show your plot corresponding to Fig. 4.16 including the fitted line. To fit the line you can use polyfit in Matlab [An example to use polyfit can be found in this file: fit.m]) )
We 02 Nov
10
31/10-02/11
Correlation Dimension +
Chaos in 2D maps +
Lyapunov Exponents.
4.7
5.1-5.2
  • Choose a project team-mate, chose a project (or project theme) and write a paragraph describing it
  • Write a code to compute the correlation dimension and compute it for the Hénon map. (show your plot of C(r) vs r and the fitted line)
  • Write a code to produce the Mandelbrot set and reproduce Fig 4.10 (hint: suppose that if |z|>2 then the orbit diverges to infinity).
  • Write a code to produce Julia sets and reproduce Fig 4.11.(a)
  • Computer experiment 6.2 (p.238) [use many different ICs p to quantify the average rate of convergence for mn ]
We 09 Nov
11
07-09/11
Lyapunov Dimension +
Fixed point theorem +
Markov partitions +
Horseshoe map +
Shadowing
5.3-5.6
  • Review: 'Regular and Chaotic Behaviour in an Extensible Pendulum'. R. Carretero-González, H.N. Núñez-Yépez and A.L. Salas-Brito. Eur. J. Phys. 15, 3 (1994) 139-148. Abstract, PDF. [Review instructions]

  • Write a code to compute Lyapunov exponents for higher dimensional maps using the Graham-Schmidt orthogonalization procedure

  • Using this code compute the Lyapunov exponents of

      a) the Hénon map,

      b) the Ikeda map, and

      c) symbiotic interaction between 3 species map:
      xn+1 = a(xn+yn+zn+1) xn (1-xn),
      yn+1 = a(xn+yn+zn+1) yn (1-yn),
      zn+1 = a(xn+yn+zn+1) zn (1-zn).
      with a=1.17. Note that there are at least 3 attractors for this value of a. Find and plot these attractors, list one set of ICs that leads to each attractor, and compute their respective Lyapunov exponents.
      Also study numerically these chaotic attractors as you vary the paramater between a=0.5 and a=1.5.
We 16 Nov
12
14-16/11
Chaotic attractors +
Forward limit sets +
Expanding maps +
Measure +
Natural Measure +
Invariant Measure +
PDF for Logistic map +
Birkhoff ergodic theorem +
Stable manifold theorem +
Homoclinic and Heteroclinic orbits +
DsTool
6.1-6.6
10.1-10.2
     
We 23 Nov
13
21-23/11
Bifurcations:
Saddle node bifurcation +
Period doubling bifurcation +
Transcritical bifurcation +
Detecting Bifurcations +
Full conditions for bifurcations
11.1-11.2
  • Ex: T5.2, 5.1, 5.2, 6.7
  • Read in depth Challenge#5
    • Ex: 10.4, 10.5
  • Ex: 11.2, 11.3
We 30 Nov
14
28-30/11
Genericity of Bifurcations +
Continuability of Bifurcations +
Bifurcations of 2D maps: area-contracting case +
Bifurcations of 2D maps: area-preserving case +
Cascades +
Feigenbaum's constant
11.3-11.5, 12.1
    Work hard on your projects!
We 07 Dec
15
05-07/12
Chaoric Time-Series +
Reconstruction +
Embedding +
Method of delays +
Taken's theorem +
Determining embedding dimension +
Manipulating time-series +
Curse of dimensionality
13 + [my notes]
    Work hard on your projects!
 
16
14/12
Project presentations  
    Wednesday Dec 14th 12:30-3:30 @ Phys-147
 

19/12
Project reports due!

 
    Project reports due!
Mon 19 Dec



Extra/Grad Credit*: extra credit is REQUIRED for graduate students.