Jimmy
here is a project that is 1/2 thinking, 1/2 computing. I have sketched
out a proposal for finding cycles of flows by extremazing stochastic path
integrals, ie a quadratic cost function integrated around a guess loop
that is equal to zero if the loop is a periodic orbit of the flow.
1. Thinking: derive a fictitious time flow which modifies the loop
while strictly decreasing the cost function
2. Thinking: implement a discretization of the loop that implements the
above fictitious time flow in a good way (keeps good
smapling of the loop ruther than clumping smapling points).
3. Programming: implement this (large) set of ODEs
4. Testing:
a) find the 2-d Duffing flow cycle, Fig 2.2 in chapter 2: Flows
Chaos: Classical and Quantum (version 10)
--- ChaosBook.org/stable/postscript.html
b) how does it work for the undamped case,
http://cns.physics.gatech.edu/ChaosBook/chapters/billiards.ps.gz
Fig 5.1?
c) find some short 3-d Roesseler cycles
http://cns.physics.gatech.edu/ChaosBook/stable/chapters/cycles.ps.gz
Fig 14.1 ...
That would be already quite good. I know c) work as both Lan and
Paskauskas have implemented such codes, and for them it convereges like
a ton of rocks.
what do you think?
Predrag
------------------------------
PS the plan for April being something like this (adopted to your own
ambitions):
Schedule of which part I intend to deliver by which date:
\begin{enumerate}
\item{\bf Thu Apr 1:}
Work out equations for ...
\\
Will integrate typical trajectories.
\item{\bf Thu Apr 8:}
Will construct the Poincar\'e section
\\
Will determine approximate symbolic dynamics for ...
\item{\bf Thu Apr 15:}
Will find periodic orbits of ...
Will use cycle expansions to compute the average of ...
\item{\bf Thu Apr 22:}
Will polishe the project to high shine ...
\item{\bf Thu Apr 29:}
Project deadline
\end{enumerate}