CHAOS, AND WHAT TO DO ABOUT IT
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Georgia Tech PHYS 7123 Fall semester 2001
Predrag Cvitanovic'
Lecture 22: 11:05-12:55 Thu Nov 1 2001 in Howey N209
Cycle expansions
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So far we have derived a plethora of periodic orbit trace formulas,
spectral determinants and zeta functions. Now we learn how to expanded
these as cycle expansions, series ordered by increasing topological
cycle length, and evaluate average quantites like escape rates. These
formulas are exact, and, when the winds are kind, highly convergent.
The pleasant surprise is that the terms in such expansions fall off
exponentially or even faster, so that a handful of shortest orbits
suffices for rather accurate estimates of asymptotic averages.
Reading:
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Classical and Quantum Chaos --- www.nbi.dk/ChaosBook
chapter 13 - Cycle expansions
Exercises:
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13.2 Escape rate for a 1-d repeller, periodic orbit theory
(please try to work through this all the way -
you might also find reading the solution
www.nbi.dk/ChaosBook/chapters/solutions.ps.gz
helpful in uderstanding the material)
rest is optional:
8.1 Escape rate for a 1-d repeller, numerically
(optional exercise - if you want to estimate the
number the cycle expansion will give you next)
10.15 Heavy pruning
(optional)
11.11 Heavy pruning - continued
(optional) work through the implementation of
pruning of blocks
10010, 101, 01001, 01101, 111, 10110
compute topological zeta function, entropy
due Thu Nov 8 at 12:55