2015/12/31 by Suddhasattwa Das, Chris B. Dock, Yoshitaka Saiki +4 · 1 citation
Physics and Astronomy · Mathematics · #math-ph #math.MP #nlin.CD #msc:37C55 #msc:37N05 #msc:37M25 #msc:37A30 #msc:37J40
paper · pdf · doi:10.1209/0295-5075/114/40005
published as Europhysics letters, 116(4), 2016 · 5 figures
arxiv created 2016/02/26 · arxiv updated 2018/01/31
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 1025 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.