Measuring quasiperiodicity

Autor: Das, Suddhasattwa, Dock, Chris B., Saiki, Yoshitaka, Salgado-Flores, Martin, Sander, Evelyn, Wu, Jin, Yorke, James A.
Rok vydání: 2015
Předmět:
Zdroj: Europhysics letters, 116(4), 2016
Druh dokumentu: Working Paper
DOI: 10.1209/0295-5075/114/40005
Popis: The Birkhoff Ergodic Theorem asserts under mild conditions that Birkhoff averages (i.e. time averages computed along a trajectory) converge to the space average. For sufficiently smooth systems, our small modification of numerical Birkhoff averages significantly speeds the convergence rate for quasiperiodic trajectories -- by a factor of $10^{25}$ for 30-digit precision arithmetic, making it a useful computational tool for autonomous dynamical systems. Many dynamical systems and especially Hamiltonian systems are a complex mix of chaotic and quasiperiodic behaviors, and chaotic trajectories near quasiperiodic points can have long near-quasiperiodic transients. Our method can help determine which initial points are in a quasiperiodic set and which are chaotic. We use our {\bf weighted Birkhoff average} to study quasiperiodic systems, to distinguishing between chaos and quasiperiodicity, and for computing rotation numbers for self-intersecting curves in the plane. Furthermore we introduce the Embedding Continuation Method which is a significantly simpler, general method for computing rotation numbers.
Comment: 5 figures
Databáze: arXiv