<- [[:chaosbook]] ====== Chapter: Discrete time dynamics ====== (ChaosBook.org blog, chapter [[http://chaosbook.org/paper.shtml#maps|Discrete time dynamics]]) --- //[[predrag.cvitanovic@physics.gatech.edu|Predrag Cvitanovic]] 2009-02-11 12:55// ===== Section: Poincaré sections ===== {{gtspring2009:pc.jpg }} If GaTech world domination is to be maintained, we need to start taking Poincaré sections of unstable manifolds. Make sure you understand chapters on [[http://chaosbook.org/paper.shtml#maps|Discrete time dynamics]], [[http://chaosbook.org/paper.shtml#stability|Local stability]] and [[http://chaosbook.org/paper.shtml#invariants|Cycle stability]]. --- //[[predrag.cvitanovic@physics.gatech.edu|Predrag Cvitanovic]] 2009-02-11 12:55// {{gtspring2009:pc.jpg }} Constructing Poincaré sections and return (or forward) maps is not such a big deal - it's just that no masters of plumbing listen to my pleas. **Jul 6 2006** I got Halcrow to give it a try: {{chaosbook:intro:returntimevsperturbation.png|}} Poincaré return time to first intersection with a Poincaré section, normal to one of the eigenvectors corresponding to the most unstable complex pair of eigenvalues of the upper branch. ~~CL~~ It was right in spirit. It was wrong in detail: Jonathan simply used a finite length, straight line segment of the linear eigenvector instead of the curved unstable manifold for initial points, and the axes scales are - as is the custom among the nonlinear graduate students - profoundly mysterious. A return map maps a curvilinear segment labeled by arclength s_n into s_{n+1}, i.e., the graph should be a square, with the same units on both axes. "Poincaré return time" is something else. He never listened to me again, so there it stands. John G's [[gtspring2009:gibson:w03|periodic orbit P47.18 in the W03 cell]] presumably sits very nicely on it. With Y. Lan I had a bit more luck. He resisted for 6 years or so, but than his wife told him that thesis should be finished this semester and he relented. His Kuramoto-Sivashinsky ("fluid dynamics" in one dimension) plots all like the usual nonlinear garbage: {{chaosbook:intro:antorb1b.png|}} projected on random coordinates (first Fourier modes) until he relented and plotted the unstable manifold of the "upper branch," {{chaosbook:intro:ant5man12.png|}} and relented even further and plotted the D_1 discrete symmetry quotiented return map: {{chaosbook:intro:ant5mmppf.png|}} And - (the mystery hidden from human eye by being written in the ChaosBook!) - very many periodic orbits followed, labeled by a ternary alphabet. So I know you can do it, if you get your mind to it. ~~DISCUSSION~~