Home Search Collections Journals About Contact us My IOPscience Measuring quasiperiodicity This content has been downloaded from IOPscience. Please scroll down to see the full text. 2016 EPL 114 40005 (http://iopscience.iop.org/0295-5075/114/4/40005) View the table of contents for this issue, or go to the journal homepage for more Download details: IP Address: 129.2.9.245 This content was downloaded on 04/08/2016 at 17:40 Please note that terms and conditions apply. May 2016 EPL, 114 (2016) 40005 www.epljournal.org doi: 10.1209/0295-5075/114/40005 Measuring quasiperiodicity Suddhasattwa Das1, Chris B. Dock2, Yoshitaka Saiki3, Martin Salgado-Flores4, Evelyn Sander5, Jin Wu6 and James A. Yorke7 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 in final form 27 May 2016 published online 16 June 2016 PACS 05.10.-a – Computational methods in statistical physics and nonlinear dynamics PACS 89.20.-a – Interdisciplinary applications of physics PACS 45.20.Jj – Lagrangian and Hamiltonian mechanics Abstract – 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 con- vergence 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 sys- tems 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. Copyright c EPLA, 2016 Introduction. – Periodicity, quasiperiodicity, and Newton’s method to find Fourier series coefficients; see for chaos are the only three types of commonly observed dy- example [2] by Jorba. namical behaviors in both deterministic models and ex- These quasiperiodic orbits occur in both Hamiltonian periments [1]. A quasiperiodic orbit of a map T lies on a and more general systems [3–14]. Luque and Villanueva [3] closed curve (or torus in higher dimensions) X, such that have published an effective method for computing rota- by a smooth change of coordinates, the dynamics of T tion numbers, see their fig. 11. On restricted three-body becomes pure rotation on the circle (respectively, torus) problems, they get 30-digit precision for rotation num- by a fixed irrational rotation number(s) ρ; that is, after bers using N ≈ 2 × 106 trajectory points while we get the change in coordinates, the map on each coordinate θi 30-digit precision with N = 20000. In this paper, they becomes θi → θi + ρi mod 1. apply their technique to rotation numbers and not other Our improved method for computing Birkhoff averages function integrals, but see also [15], where they used a for quasiperiodic trajectories enables the computation of slower convergence method for Fourier series. More detail rotation numbers, which are key parameters of these or- about our results here can be found in [16] (numerical) bits. It also allows computation of the torus on which and Corollary 2.1 from [17] (theoretical). We should note an orbit lies and of the change of coordinates that con- that the Birkhoff approach (and ours) assumes we have a verts the dynamics to a pure rotation. Our time series trajectory on the (quasiperiodic) set. data is not appropriate for an FFT, but there is a stan- Distinguishing between quasiperiodic and chaotic be- dard way of computing such a change of coordinates using havior in borderline cases is a difficult and important 40005-p1 Suddhasattwa Das et al. current topic of research for both models and experiments A more general class of such C∞ weight functions for in physics [18–21] and biology [22,23], and finding good p ≥ 1isw[p](t):=exp(−[t(1 − t)]−p)fort ∈ (0, 1) and numerical methods is a subject of active study [24]. The = 0 elsewhere. Our examples use w = w[1] here, but coexistence of chaos and quasiperiodicity arbitrarily close w[2] is even faster when requiring 30-digit precision. It to each other in a fractal pattern makes this detection a is no faster when requiring 15-digit precision. The above difficult problem. Recently proposed methods [25,26] suc- constant Cm depends on i) w(t) and its first m derivatives; cessfully distinguish between different invariant sets, but ii) the function f(t); and iii) the rotation number(s) of the methods suffer from extremely slow convergence due to the quasiperiodic trajectory or more precisely, the small their reliance on the use of Birkhoff averages. By combin- divisors arising out of the rotation vector. We do not have ing [25,26] with our method of weighted Birkhoff averages, a sharp estimate on the size of the term Cm. we are able to distinguish between chaos and quasiperiod- As a result of this speed, we are able to obtain high icity with excellent accuracy, even in cases in which other precision values for fdμ with a short trajectory and rel- methods of chaos detection such as the method of Lya- atively low computational cost, largely independent of the punov exponents, fail to give decisive answers. choice of the C∞ function f. We get high-accuracy re- sults for rotation numbers and change of coordinates to a The Birkhoff average. – For a map T ,letx = T nx n pure rotation for the Standard Map and the three-body be either a chaotic or a quasiperiodic trajectory. The problem. For a higher-dimensional example and further Birkhoff average of a function f along the trajectory is details, see [16]. N−1 In creating WBN , we were motivated by “apodization” n BN (f)(x):= (1/N )f(T (x)). (1) in optics (especially astronomy and photography), where n=0 diffraction that is caused by edge effects of lenses or mir- Under mild hypotheses the Birkhoff Ergodic Theorem con- rors can be greatly decreased. Our weighting method is cludes that BN (f)(x) → fdμ as N →∞where μ is an reminiscent of both Hamming windows and Hann (or Han- invariant probability measure for the trajectory’s closure. ning) filters for a Fourier transform on small windows (see This relationship between the time and space averages is for example, [27–29]. The analogue of w usually has only incredibly powerful, allowing computation of fdμ when- a couple derivatives = 0 at the end points t =0, 1andso ever a time series is the only information available. How- convergence rate is only slightly better than the conver- ever, the convergence of the Birkhoff average is slow, with gence rate of the standard Birkhoff method [16]. an error of at least the order N −1 for a length N trajectory Testing for chaos. – WB also provides a quanti- in the quasiperiodic case. N tative method of distinguishing quasiperiodic trajectories Weighted Birkhoff (WB ) average. Instead of using N from chaotic trajectories. Along a trajectory xn,wecan Birkhoff’s uniform weighting of f(x ), our average of these n compare the value of WBN (f) along the first N iter- values gives very small weights to the terms f(xn) when n − ates with WBN (f) along the second N iterates, i.e. we is near 0 or N. Set w(t):=exp(−[t(1−t)] 1)fort ∈ (0, 1) N consider ΔN =WBN (f)(x) − WBN (f)(T (x)). For a and = 0 elsewhere. Define the Weighted Birkhoff average quasiperiodic orbit, we expect |ΔN | to be very small. To (WB )off as follows: N measure how small |ΔN | is, we can count the number of N−1 zeros after the decimal point by defining WBN (f)(x):= wˆn,N f(xn), (2) zerosN (f)(x)=− log |ΔN |. (3) n=0 10 −1/2 N−1 If the orbit is chaotic then |ΔN |∼N or slower, zerosN wherew ˆn,N = w(n/N)/Σ w(j/N). WBN (f) has the j=0 is small. Whereas if it is quasiperiodic, both WB f(x) same limit f as the Birkhoff average but on quasiperiodic N and WB (f(T N (x))) have super convergence to fdμ trajectories WB (f) converges to that limit with 30-digit N N and so Δ has super convergence to 0, implying that precision faster than B by a factor of about 1025. (There N N zeros is large. For example, see fig. 1. To check the ac- is no increase in convergence rate for chaotic trajectories.) N curacy of our method, we tested 12, 086 initial conditions Intuitively, the improvement arises since the weight func- on the diagonal {x = y} for the Standard Map (eq. (4)). tion w vanishes at the ends, and thus gets rid of edge We found that 99.8 per cent of the initial conditions for effects. We have proved [17] that if (xn) is a quasiperiodic which zerosN > 18 for N = 20000 are in fact quasiperiodic trajectory and f and T are infinitely differentiable (i.e.