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}