Issue |
EPL
Volume 114, Number 4, May 2016
|
|
---|---|---|
Article Number | 40005 | |
Number of page(s) | 6 | |
Section | General | |
DOI | https://doi.org/10.1209/0295-5075/114/40005 | |
Published online | 16 June 2016 |
Measuring quasiperiodicity
1 Department of Mathematics, University of Maryland - College Park, MD, USA
2 University of California, Berkeley - Berkeley, CA, USA
3 Graduate School of Commerce and Management, Hitotsubashi University - 2-1 Naka, Kunitachi, Tokyo, Japan
4 College of William and Mary - Williamsburg, VA, USA
5 Department of Mathematical Sciences, George Mason University - Fairfax, VA, USA
6 University of Maryland - College Park, MD, USA
7 Department of Mathematics, Physics, and IPST, University of Maryland - College Park, MD, USA
Received: 22 December 2015
Accepted: 27 May 2016
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 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.
PACS: 05.10.-a – Computational methods in statistical physics and nonlinear dynamics / 89.20.-a – Interdisciplinary applications of physics / 45.20.Jj – Lagrangian and Hamiltonian mechanics
© EPLA, 2016
Current usage metrics show cumulative count of Article Views (full-text article views including HTML views, PDF and ePub downloads, according to the available data) and Abstracts Views on Vision4Press platform.
Data correspond to usage on the plateform after 2015. The current usage metrics is available 48-96 hours after online publication and is updated daily on week days.
Initial download of the metrics may take a while.