Volume 114, Number 4, May 2016
|Number of page(s)||6|
|Published online||16 June 2016|
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
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