Assignment schedule:

THESE WEBPAGES DESCRIBE HOW THIS COURSE WILL BE RUN.
Browse through all the options in the menu above. Visit this webpage regularly for updates

Homework: Typically there will be one homework per week. The HW will be posted here every Tuesday (at the latest) and is due the following Thursday night and must be submitted in Gradescope. In some occasions the HW will be posted earlier so you are encouraged to start on it earlier. Late HW 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
27-29/08
Introduction (pptx presentation) +

1D maps:
Cobweb plots +
1.1-1.2
[Lec_2.pdf]
  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/explore all instructions in Matlab (follow the whole tutorial) and submit to me by email a diary of it.
    To do the diary:
    • Write "diary LastName_M538.txt" as the first instruction.
    • Type/explore the tutorial. Everything you key will be saved.
    • Type "diary OFF". This will close the diary and save the file in your current directory (nothing is saved to the file until you close the diary).
    • If you want to do the diary in several seatings, just open different diaries every time using "diary LastName_M538_A.txt", "diary LastName_M538_B.txt", etc...
    • Email me your diary(ies).
 HW#01:
  • 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 (turn in a sample code as well) and explain your results.
Thu 05 Sep
02
03-05/09
1D maps:
Stability of fixed points +
Basins of attraction +
Periodic orbits +
Logistic map family
1.3, 1.4
[Lec_3.pdf]
[Lec_4.pdf]
  HW#02:
  • Ex T1.3, T1.4 (p. 12-13).
  • Ex 1.1, 1.2, 1.3 (p. 36).
  • 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.
Thu 12 Sep
03
10-12/09
1D maps:
The logistic map +
Sensitive dependence to ICs +
Itineraries, Symbolic Dynamics

1.5-1.8
[Lec_5.pdf]
[Lec_6.pdf]
  HW#03:
  • Reproduce Figs. 1.6 and Fig 1.7 (p. 19-20 of our textbook) [include a copy of your code!].
  • Ex T1.5, T1.7, T1.8, T1.9 (p. 14-24).
  • Ex 1.5, 1.7, 1.10, 1.12 (in (b) use cobweb) (p. 36-37).
  • Review: 'Simple mathematical models with very complicated dynamics', R. May, Nature 261 (1976) p. 459. [PDF]
    [Review instructions]
  • 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]
Thu 19 Sep
04
17-19/09
2D maps:
Poincaré sections +
Hénon map +
Sink, sources and saddles +
Linear maps +
Real evals +
Repeated evals
2.1-2.3
[Lec_7.pdf]
[Lec_8.pdf]
  HW#04:
  • Read/Study eigenvalue notes from "Linear Algebra and Diff. Eqns. Using Matlab" by Golubitsky and Dellnitz.
    [2x2 matrices] [NxN matrices]
  • Write a summary of all important formulas and definitions from these eigenvalue notes.
  • Reproduce Fig 5 of the article "Chaos in the cubic mapping" (see above, [PDF]).
    Note: the procedure on how to produce bifurcation diagrams, in addition of what I explained in class, it is briefly explained in the first paragraph in p. 18 of our textbook.
  • Reproduce Fig. 2.3 (p. 51 of our textbook). Note that Fig. 2.3.b is with b=-0.3 and a=1.5 (and not a=1.4)
  • Reproduce Fig. 2.10 (p. 61 of our textbook).
  • Reproduce Fig. 2.11 (p. 61 of our textbook).
Thu 26 Sep
05
24-26/09
2D maps:
Complex evals +
Coordinate Changes +
Nonlinear maps: Jacobian +
Periodic orbits
2.4-2.5
[Lec_9.pdf]
[Lec_10.pdf]
[Lec_11.pdf]
  HW#05:
  • Ex 2.3, 2.5, 2.6
  • Ex T2.2, T2.3, T2.4, T2.5, T2.7
  • Study for Midterm#1
Thu 03 Oct
06
01-03/10
2D maps:
Bifurcation diagram of Hénon map +
Stable and Unstable manifolds +
MT#1
2.6-2.7
[Lec_12.pdf]
[Lec_13.pdf]
  
MT#1
 03/10
MIDTERM#1, Thursday October 3rd
Chaps 1+2
    MT#1  
07
08-10/10
Matrix times circle equals ellipse +

Chaos:
Chaotic orbits +
Lyapunov exponents +
Tent map
3.1-3.3
[Lec_14.pdf]
  • Take home MT#1.
Tue 08 Oct
08
15-17/10
Chaos:
Conjugacy and the logistic map +
3.3
[Lec_15.pdf]
  HW#06:
  • For the logistic map f(x)=ax(1-x), write a program to compute the Lyapunov exponent and reproduce Fig. 6.3. Also plot (just below) the bifurcation diagram for the same a-window and compare/discuss both plots. Elaborate!
  • 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
  • Do not forget to include in GS a version of your Lyapunov exponents code (but do NOT email it to me).
Thu 24 Oct
09
22-24/10
Dense orbits +
Transition graphs and fixed points +
Basins of attraction

Fractals:
Cantor sets +
Iterated Function Systems
3.4-3.5, 4.1-4.2
[Lec_16.pdf]
  HW#07:
  • 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)
  • 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. Important: 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]. To count the boxes use the histogram command histcounts2 in Matlab [An example to use the histcounts2 command to count boxes can be found in this file: henon_histogram.m].
  • Write a code to compute the correlation dimension and compute it for the Hénon map. Important: show your plot of C(r) vs r and the fitted line.
  • Extra/Grad Credit*: Computer experiment 6.2 (p.238) [use many different ICs p to quantify the average rate of convergence for mn ]
  • For this HW please do not forget to include a copy of your Mandelbrot + Julia + box counting + correlation dimension codes with your GS submission (do NOT email me the codes).
Tue 12 Nov
10
29-31/10
Deterministic Fractals +
Mandelbrot set +
Julia sets
4.3-4.4
[Lec_17.pdf]
[Lec_18.pdf]
    
11
05-07/11
Fractal dimension +
Box Counting Dimension
Correlation Dimension +

Chaos in 2D maps:
Lyapunov Exponents
4.4-4.7
5.1
[Lec_19.pdf]
[Lec_20.pdf]
  HW#08:
  • Choose project team-mate(s), chose a project (or project theme) and write a paragraph describing it

  • 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]

  • Using the posted code [lyapunov_QR_2D.m], compute the Lyapunov exponents for for the following maps and comments on your results.

      a) the Hénon map,

      b) the Ikeda map (use the parameter values given in p. 202), and

      c) the 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.
        (i) Find and plot these attractors (they should look like this: pic1, pic2, pic3).
        (ii) List one set of ICs that leads to each attractor and compute their respective Lyapunov exponents.
        (iii) Describe what happens to these chaotic attractors as you vary the paramater between a=0.5 and a=1.5?
      Note: for this map you'll need to extend the posted code from 2D to 3D.

  • Compute the Lyapunov exponents of the Hénon map using the eigenvalues of (J JT) where J is the Jacobian of the n-th iterate of the map (remember that J = Jn ... J2 J1 ). Do this for n=5,10,15,20,25,30,35,40,45,50,... and compare the results that you get for the different values of n with the result you got in a) of the previous problem. Please comment/explain/elaborate!

Thu 21 Nov
12
12-14/11
Numerical Computation of Lyap. Exp. +
Lyapunov Dimension
5.2-5.3
[Lec_21.pdf]
    
13
19-21/11
Fixed point theorem +
Markov partitions +
Horseshoe map +
Shadowing +

Chaotic attractors:
Forward limit sets +
Chaotic attractors
5.4, 5.6, 6.1, 6.2
[Lec_22.pdf]
[Lec_23.pdf]
  • Work hard on your projects! About 10hrs/week !!!
  
14
26/11
Chaotic attractors of expanding maps +
Measure +
Natural Measure +
Invariant Measure +

Thanksgiving break
6.3-6.6
[Lec_24.pdf]
  HW#09:
  • Ex: 6.5, 6.7, 6.8
  • Work hard on your projects! About 10hrs/week !!!
Thu 05 Dec
15
03-05/12
Invariant Measure for piece-wise linear expanding maps +
PDF for Logistic map +
Birkhoff ergodic theorem

Stable Manifolds and Crises:
Stable manifold theorem +
Homoclinic and Heteroclinic orbits +

Bifurcations:
Saddle-node bif. +
Period-doubling bif. +
Transcrtital bif. +
Pitchfork bif. +
Determining bifurcations
6.6, 10.1, 10.2, 11.1, 11.2
[Lec_25.pdf]
[Lec_26.pdf]
  
  • Work hard on your projects! About 10hrs/week !!!
  
16
10/12
Bifurcation diagrams +
Continuability +
Bifurcations of 1D maps+
Bifurcations in the plane +
Area presenving case +
Neimark-Sacker bifurcation +
Full conditions for bifurcations
11.3-11.6
[Lec_27.pdf]
  HW#10:
  • Ex: 11.2, 11.3&, 11.7, 11.11
    • Notes:
    • For the above exercises use the conditions on the derivatives to determine bifurcations.
    • &Be careful in Ex. 11.3 as the conditions using partial derivatives will not work. Explain! Also, the answer in the book says that the period-doubling bifurcation happens at a=-1 when it actually happens at a=+1.

  • Work hard on your projects! About 10hrs/week !!!
Thu 12 Dec
17
17/12
Project presentations
Instructions for project reports/presentations 		    Tuesday Dec 17th 12:45-3:15 @ GMCS-305
  Instructions for project reports/presentations
 

21/12
Project reports due!

 
Sat 21 Dec



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