INTRODUCTION TO NONLINEAR DYNAMICS AND CHAOS
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Georgia Tech PHYS 4426/6268 Spring semester 2007
Predrag Cvitanovic'
Problem set 6: Lozi map due Tue Feb 20 2007
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Feb 18 2007
NOTE added comments, in response to several emails
now reaching lenght of a novel.
Problem 5.18A [not in the book]
NOTE [equtation number, some text edited here]:
(a) express the Lozi map (5.44) coordinates x,y, parameters (a,b)
in terms of the "roof" map (5.37) x,v, parameters (E,a).
(b) Compute fix points of the Lozi map (5.44), check whether
your formulas agree with (5.38) after parameters (a,b)
are expressed in terms of parameters (E,a).
(c) Compute eigenvalues, eigenvectors of the fixed points.
NOTE added: do not trust in detail anything that was
written on the blackboard in lectures - trust yourself.
(d) Plot Lozi fixed points, their eigenvectors.
Problem 5.18B [not in the book]
Program the Lozi map (5.54) and reproduce
the Lozi map version of Tel and Gruiz Figure 5.28
(a) for Lozi parameters corresponding to "roof" (a,E)=(1.77,0.7).
Plot the long time trajectory in the (x_n,x_{n+1}) plane,
rather than (x_n,y_n), it's a more convenient representation.
NOTE added: plot the attractor by iteration, as Fig 5.28,
not by computing straight-line segments as Fig 5.29, and
as discussed in the lecture.
That would be too much work as a part of a problem set.
(b) NOTE [text edited here]:
Enter into the same computer plot the fixed points,
eigenvectors from the Problem 5.18A(d). Does the unstable
eigenvector align with the attractor?
Problem 5.19
NOTE. Text of advice edited a bit:
- Do not magnify by 30, 300 as in the book solution, but
by inverse powers of the stable eigenvalue of H_{+}:
1/\Lambda_{-}^n as defined in (5.29) and a few lines below
Try overlaying the images to see how self-similar they are.
You know this is a good idea from the discussion of
homoclinic tangles: in a linearized neighborhood multiplication
by 1/\Lambda_{-} maps a homoclinic point h_n into the
preceeding point h_{n-1}, so along the unstable manifold you
are guaranteed exact self-similarity for this family of points.
As \Lambda_{-} is negative, for magnifications by odd powers
the neighborhood will be a mirror image of the even ones.
- Generating such figures requires very long runs.
You can estimate a number of trajectory points needed:
x_n they are randomly distributed
in an interval of approximate width 3, larger enlargement is
of width 0.025 and 1000's of points need to land in it.
So you probably do not want to save the trajectory points as
a numerical data file, but rather plot points directly into
a graphics file.
----- graduate (extra points for undergrads):
Problem 5.12
----- bonus points for all:
Problem 5.19 [Thinking is extra price version]
Explain why Fig. P.16 is NOT self-similar.
NOTE added: [self-similar means that a magnified detail
has all the same pieces as the entire fractal, as in all
illustrations in chapter 2]
Hint: map forward in time any of the unstable manifold turnbacks
at x=0 (critical points), observe how they visit the neighborhood
of H+ fixed point.
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append a printout of your program to the problem set, or - if
you would like to have your program posted on the course homepage -
email the program to
Jonathan Halcrow gte899j [ at ] mail.gatech.edu
www.cns.gatech.edu/~predrag/phys4267
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Feb 15 and 18 2007 -- Predrag.Cvitanovic@physics.gatech.edu