From Trajectories to the Ergodic Partition - An Algorithm

From Trajectories to the Ergodic PartitionAn AlgorithmMarko Budisic (presenter)mbudisic@engineering.ucsb.eduDr. Igor Mezicmezic@engineering.ucsb.eduUniversity of California, Santa BarbaraDynamics Days, Jan 3{6, 2008Knoxville, TNApproved for Public Release, Distribution Unlimited,DARPA, 02/11/2008Motivation and purposeTrajectory plot Approx. of ergodic partitionλ = 0.3; R = 100; N = 5000 101980.7576)0.5540.25320 10 0.25 0.5 0.75 1θi(mod1)Target GoalMeasure-preserving map on a Quick, coarse partition of phase nite/periodic domain. space.Approved for Public Release, Distribution Unlimited,DARPA, 02/11/2008J (mod1)i15 min talk in one slideCore idea How can we do it?Ergodic subsets { dynamical "Concatenate" trajectories usingatoms in phase space. data clustering methods.Why do we care? Does our solution measure up?Analysis { mapping out Yes.phase space Fast ( minutes) two-stepalgorithm.Design { easier toPartition corresponds to dynamics inexploit naturalknown problems.dynamics ofsystemApproved for Public Release, Distribution Unlimited,DARPA, 02/11/2008Theoretical frameworkBirkho ’s ergodic theoremFor system T :M!M, ifXM is an ergodic subset, then for18x2X;8f2 L (M)temporal averagespatial averagez }| {z }| {Z n h iX1 k f = f (x)d(x) = lim f T (x) = f (x);n!1 nk=0XLevel sets of f ! invariant Grouping criterionpartitionPf 1f (x) = f (y);8f2 LErgodic partition +WP := P1E ff2L x; y2XApproved for Public ...