Testing for Chaos in Type-I ELM Dynamics on JET with the ILW. Fabio Pisano

Transcription

1 Testing for Chaos in Type-I ELM Dynamics on JET with the ILW Fabio Pisano

2 ACKNOWLEDGMENTS B. Cannas 1, A. Fanni 1, A. Murari 2, F. Pisano 1 and JET Contributors* EUROfusion Consortium, JET, Culham Science Centre, Abingdon, OX14 3DB, UK 1 Department of Electrical and Electronic Engineering - University of Cagliari, Italy 2 Consorzio RFX-Associazione EURATOM ENEA per la Fusione, I Padova, Italy * See the Appendix of F. Romanelli et al., Proceedings of the 25th IAEA Fusion Energy Conference 2014, Saint Petersburg, Russia F. Pisano 1 st IAEA Technical Meeting on Fusion Data Processing, Validation and Analysis Nice 1-3 June Page 2

3 Introduction Surrogate Data Linear Surrogate Data PseudoPeriodic Surrogate Data Autocorrelation and TDMI State Space Reconstruction Correlation Dimension Sensitivity to Initial Conditions Weak Chaos and Intermittency Attractors Conclusions F. Pisano 1 st IAEA Technical Meeting on Fusion Data Processing, Validation and Analysis Nice 1-3 June Page 3

4 Apparently aperiodic and hardly predictable behaviour of ELMs Deep impact of the ILW at JET on the physics of ELMs in JET. o o Significant change of the ELM dynamics observed for spontaneous type-i ELMs ELM duration much longer for the ILW than in similar cases with a CW. An analysis made on CW campaigns highlighted a weak chaotic and pseudoperiodic behaviour on ELM dynamics o o o Coexistence of a periodic component and of a stochastic one Behaviour typical of intermittent systems Connection with intermittence theory F. Pisano 1 st IAEA Technical Meeting on Fusion Data Processing, Validation and Analysis Nice 1-3 June Page 4

5 BASE o Fixed experimental conditions for stationarity o CW o ILW 19 shots C21-C27b (9 th June rd October 2010) D α signals from outer divertor (S3AD/AD35) 28 shots C29-C30 (8 th March th June 2012) Total outer divertor Be II photon flux (EDG8/TBEO) Total outer divertor D photon flux (EDG8/TDAO) Consistent results obtained on ILW with Be II have been found also using D F. Pisano 1 st IAEA Technical Meeting on Fusion Data Processing, Validation and Analysis Nice 1-3 June Page 5

6 CW ILW F. Pisano 1 st IAEA Technical Meeting on Fusion Data Processing, Validation and Analysis Nice 1-3 June Page 6

7 The objective of the technique of surrogate data is to create a comparison test set created from the data o Surrogates keep some properties of the original data set while destroying the others Two types of surrogate data used o Linear surrogates nonlinearity Maintain the same linear properties of the original time series while destroying its nonlinear structure Same power spectrum and probability distribution o Pseudoperiodic surrogates pseudoperiodicity Preserve the potential pseudoperiodicity of the original time series Pseudoperiodicity = periodicity with observational and/or dynamical noise F. Pisano 1 st IAEA Technical Meeting on Fusion Data Processing, Validation and Analysis Nice 1-3 June Page 7

8 LINEAR Null hypothesis: data generated by a linear Gaussian process, distorted by a nonlinear static invertible function [Schreiber 1999] Nonlinear invertible function y n f Linear Gaussian process xn xn ai xn i M, b Same probability distribution and power spectrum as the original series If nonlinear measures between original time series and surrogates are Similar Different Then: i 1 N i 0 Null hypothesis cannot be rejected y n comes from a stochastic process i n i Null hypothesis can be rejected y n comes from a nonlinear deterministic process F. Pisano 1 st IAEA Technical Meeting on Fusion Data Processing, Validation and Analysis Nice 1-3 June Page 8

9 LINEAR F. Pisano 1 st IAEA Technical Meeting on Fusion Data Processing, Validation and Analysis Nice 1-3 June Page 9

10 Null hypothesis: data pseudoperiodic with noise components identically distributed and uncorrelated [Luo 2005] y n 2 y y, 0 1 n 1 n 0 0 sufficiently large in order to uncorrelate the noise components If: y n pseudoperiodic with observational or low dynamical noise y n pseudoperiodic with strong dynamical noise y n chaotic Then: y n pseudoperiodic with the same correlation dimension Unreliable results y n with a different correlation dimension F. Pisano 1 st IAEA Technical Meeting on Fusion Data Processing, Validation and Analysis Nice 1-3 June Page 10

11 Null hypothesis: data pseudoperiodic with noise components identically distributed and uncorrelated [Luo 2005] y n 2 y y, 0 1 n 1 n 0 0 sufficiently large in order to uncorrelate the noise components If: Similar correlation dimensions Different correlation dimensions Then: Null hypothesis cannot be rejected y n pseudoperiodic Further tests needed y n chaotic or pseudoperiodic with strong dynamical noise F. Pisano 1 st IAEA Technical Meeting on Fusion Data Processing, Validation and Analysis Nice 1-3 June Page 11

12 F. Pisano 1 st IAEA Technical Meeting on Fusion Data Processing, Validation and Analysis Nice 1-3 June Page 12

13 Correlation of a time series with its own past and future values Autocorrelation Function, R Linear dependence of a variable with itself at two points at distance in the time series TDMI, Time Delayed Mutual Information, I Nonlinear version of autocorrelation function Measure of the average amount of shared information gained on the time series from the time series delayed of a time lag F. Pisano 1 st IAEA Technical Meeting on Fusion Data Processing, Validation and Analysis Nice 1-3 June Page 13

14 CW ILW Linear surrogates behave similarly to the original series for construction Deviation of pseudoperiodic surrogates close to 0 Linear surrogates collapse immediately (nonlinearity) Pseudoperiodic surrogates behave similarly to the original series (pseudoperiodicity) F. Pisano 1 st IAEA Technical Meeting on Fusion Data Processing, Validation and Analysis Nice 1-3 June Page 14

15 Takens embedding theorem dx( t) dt f ( x) Dynamical system * m if m > 2D y n D fractal dimension time delay embedding dimension y( x( n t) Observable time series Attractor structure y -5 [Takens 1981] 0 n [ y *,..., y *, y n ( m 1) n n Embedding space )] F. Pisano 1 st IAEA Technical Meeting on Fusion Data Processing, Validation and Analysis Nice 1-3 June Page 15

16 Time delay, * o Coordinate components slightly correlated while still being close to one another If is small compared to the internal time scales of the system, successive elements of the delay vectors are strongly correlated. If is very large, successive elements are already almost independent. CW ILW * = 1 F. Pisano 1 st IAEA Technical Meeting on Fusion Data Processing, Validation and Analysis Nice 1-3 June Page 16

17 Embedding dimension, m o False Nearest Neighbours (FNNs) algorithm A suitable embedding dimension is found when most of the nearest neighbours do not move apart significantly in the next higher dimensional embedding (FNN<30%) CW ILW m 4 7 m 3 4 F. Pisano 1 st IAEA Technical Meeting on Fusion Data Processing, Validation and Analysis Nice 1-3 June Page 17

18 Correlation dimension, D 2 o Computed by the correlation sum which represents the probability of finding two points in the embedding space at distance lower than ε o As ε 0, when m > D 2 Cm,ε D 2 Check convergence with m CW ILW F. Pisano 1 st IAEA Technical Meeting on Fusion Data Processing, Validation and Analysis Nice 1-3 June Page 18

19 CW ILW No saturation high dimensionality due to noise Clear difference between time series and linear surrogates nonlinearity Small difference between time series and p.p. surrogates pseudoperiodicity Smaller difference between time series and linear surrogates noise? Small difference between time series and p.p. surrogates pseudoperiodicity F. Pisano 1 st IAEA Technical Meeting on Fusion Data Processing, Validation and Analysis Nice 1-3 June Page 19

20 Lyapunov Exponents (LEs) o Identify the long-term qualitative behaviour of a dynamical system o Measure the rate at which nearby orbits converge or diverge. o Quantify the sensibility of the system to initial conditions o o In a chaotic system, the trajectories, on average, diverge at an exponential rate characterized by the Maximum Lyapunov Exponent (MLE) It is possible to reconstruct the MLE from experimental data [Rosenstein 1993] o Robust, consistent and unbiased estimator F. Pisano 1 st IAEA Technical Meeting on Fusion Data Processing, Validation and Analysis Nice 1-3 June Page 20

21 CW ILW Lack of exponential divergence, MLE 0 PSIC (Power law Sensitivity to Initial Condition) weak chaos Clear difference between time series and linear surrogates nonlinearity Small difference between time series and p.p. surrogates pseudoperiodicity F. Pisano 1 st IAEA Technical Meeting on Fusion Data Processing, Validation and Analysis Nice 1-3 June Page 21

22 Introduced by Pomeau and Manneville to describe the intermittency routes to chaos Prototype models to study the behavior of systems exhibiting weak chaos F. Pisano 1 st IAEA Technical Meeting on Fusion Data Processing, Validation and Analysis Nice 1-3 June Page 22

23 Embedding using high embedding dimension Principal Component Analysis (PCA) to obtain the first three principal components CW ILW 92.6% variance 99.1% variance F. Pisano 1 st IAEA Technical Meeting on Fusion Data Processing, Validation and Analysis Nice 1-3 June Page 23

24 Surrogates analysis o No compatibility with the assumption of a Gaussian random process with only linear correlations, distorted by a static nonlinear measurement function o Compatibility with the assumption of pseudoperiodicity, periodicity distorted by dynamical noise No particular difference between the dynamic behaviour of ELMs in CW and ILW o Weak chaos confirmed Interpreting ELM time series from the standpoint of PSIC and intermittency opens a new perspective in the field of ELM modelization o Use of low dimensional nonlinear maps F. Pisano 1 st IAEA Technical Meeting on Fusion Data Processing, Validation and Analysis Nice 1-3 June Page 24

25 THANKS FOR YOUR ATTENTION! QUESTIONS?