Analysis of chaotic multi-variate time-series from spatio-temporal dynamical systems Odd-Halvdan Sakse 0rstavik Thesis submitted for the degree of Doctor of Philosophy Centre for Nonlinear Dynamics and its Applications University College London 19.08.99 ProQuest Number: U642839 All rights reserved INFORMATION TO ALL USERS The quality of this reproduction is dependent upon the quality of the copy submitted. In the unlikely event that the author did not send a complete manuscript and there are missing pages, these will be noted. Also, if material had to be removed, a note will indicate the deletion. uest. ProQuest U642839 Published by ProQuest LLC(2016). Copyright of the Dissertation is held by the Author. All rights reserved. This work is protected against unauthorized copying under Title 17, United States Code. Microform Edition © ProQuest LLC. ProQuest LLC 789 East Eisenhower Parkway P.O. Box 1346 Ann Arbor, Ml 48106-1346 Acknowledgements This work would have not been possible without the guidance and encourage­ ment given to me by my supervisor Dr. Jaroslav Stark. Thank you very much. Thanks are also due to Dr. Ricardo Carretero-Gonzalez which I worked closely with on most of my thesis. A lot of collaboration was done with Professor David Broomhead and Dr. Jerry Huke at University of Manchester Institute of Science and Technology from which I benefited greatly. I also enjoyed discussions with Dr. Thomas Schreiber. I would also like to thank the Professor Michael Thompson and Professor Steven Bishop and all my fellow students at the Centre for Nonlinear Dynamics and its Applications for creating such a pleasant environment while I undertook this work. Last, but not least, I would like to give a huge thank you to Sarah for putting up with me throughout the last 3 years. Abstract This work concerns the analysis of chaotic multi-variate time-series from spatio- temporal dynamical systems (STS). Such systems can be thought of as consisting of a collection of sub-systems at different spatial locations coupled together into one large system. These arise in many applications throughout science and en­ gineering including most types of fluid flow, pattern formation in chemical and biological systems, dynamics of ecosystems, road traffic, vibration of structures such as beams, plates and shells, and many others. In many situations there is a desire to analyse data from STS in situations where little is known about the system generating the data. In particular one may have no idea of the system’s structure, or even its state space. It has until now been an open question how to characterise, control and predict the future evolution of STS in these circum­ stances. To answer these questions this thesis builds on the chaotic time-series analysis framework that has been successfully developed for the analysis of low­ dimensional systems. Coupled map lattices (CML) are used as model systems since these feature many of the characteristics of STS. Several new results that apply to spatio-temporal systems are presented and can be summarised as follows. By using a mix of temporal and spatial embedding techniques one is able to carry out reconstruction and cross-prediction on a time-series generated by a CML and the results show that spatio-temporal delay reconstructions give better predictability than standard methods using either time delays or spatial delays only. A framework for embedding spatio-temporal systems is proposed. Results also show that by using spatio-temporal embedding techniques with local observations one cannot detect the presence of spatial extent in CML’s thus suggesting the impossibility of reconstructing the whole system from localised in­ formation. New methods for calculating Lyapunov spectra for STS, and for extracting related quantities such as KS entropy density and Lyapunov dimension density, have been developed both for the case where the underlying dynamics is known and directly from time-series. CONTENTS Contents 1 Introduction 14 1.1 Thesis Outline .............................................................................................. 16 2 Background and theory 19 2.1 Chaotic time-series analysis .................................................................... 19 2.1.1 Reconstruction and embedding .............................................. 20 2.1.2 Local predictors .............................................................................. 21 2.1.3 Lyapunov spectrum and related qu a n titie s ............................... 22 2.1.4 Lyapunov spectrum from tim e-series ....................................... 25 2.2 Spatio-temporal dynamical system s ....................................................... 27 2.2.1 Coupled map la ttic e s .................................................................... 28 2.3 S u m m a ry .................................................................................................... 28 3 Reconstruction and cross prediction using spatio-temporal em­ bedding techniques 29 3.1 Spatio-temporal embeddings .................................................................... 32 3.2 Cross prediction .......................................................................................... 34 3.3 Results of predictions.................................................................................. 34 3.4 Scaling Laws .................................................................................................. 43 3.5 D iscussion .................................................................................................... 46 4 Truncated lattices and the thermodynamic limit 50 4.1 R esults.......................................................................................................... 52 4.2 D iscussion .................................................................................................... 58 CONTENTS 5 Interleaving and rescaling of the Lyapunov spectrum 60 5.1 Interleaving and rescaling for homogeneous s ta te s ...................... 63 5.2 Interleaving and rescaling for coupled logistic m a p s ................... 71 5.3 Estimation of quantities derived from the Lyapunov spectrum . 80 5.4 More general extended dynamical system s ................................... 83 5.4.1 Chaotic neural n etw o rk s ............................................................... 83 5.4.2 Two-dimensional logistic la ttic e .................................................. 85 5.4.3 Host-parasitoid sy ste m .................................................................. 92 5.5 D iscussion ............................................................................................. 97 6 Estimation of intensive quantities in spatio-temporal systems from time-series 99 6.1 Lyapunov spectrum and related densities ........................................100 6.2 Sub-system Lyapunov spectrum from time-series ...........................104 6.3 Numerical results for the estimation of the LS from time-series . 106 6.4 D iscussion .............................................................................................. 118 7 Conclusions 119 LIST OF FIGURES List of Figures 3.1 Time history of the first 1000 points of the sample set at j = 0 ......................35 3.2 Time delay reconstruction at one node .......................................................... 36 3.3 One-step prediction error (E) for increasing number of nearest neigh­ bours {k = 1 ,..., 3000) and for different choices of delay maps (d„, ds). The top three curves are for usual time delay map giving a minimum error for = 4. When we instead use dn = ds = 2 i.e. the same embed­ ding dimension size but including spatial information the predictions are substantially better. We see that the error is minimized for small values of k. The line vl is a fit of the scaling law in eq.(3.8) .................................. 37 3.4 One-step prediction error for different reconstruction designs using 50 nearest neighbours .......................................................................................... 38 3.5 One-step prediction error for different reconstruction designs using quadratic fit and 150 nearest neighbours ....................................................................... 39 3.6 Predicted (o) and actual (—) values for the first 150 points in the test set, = 50,p = 2. (-h) denotes predicted minus actual values ........................40 3.7 Prediction error as we increase the spatial delay. Here we have plotted the cases (d„, ds) = (1,2), (2,2), (3,2). We used A; = 50 neighbours. The horizontal lines depict the error when we have reconstruction in time only (the top one is for dn = I, the middle is dn = 2 and the bottom is for = 3)....................................................................................................... 41 3.8 Prediction error for j = 1 — 6, T = 0 — 5 using dn = 2,ds = 2,k = 19. 41 3.9 Absolute value of space-time two-point correlations. Actually we have plotted 1 — |C| against Aj and r .................................................................. 42 LIST OF FIGURES 3.10 One-step prediction error for increasing Ng and for different choices of {dn, dg). 20 nearest neighbours were used in all cases. The imposed lines all have slope -0.11.......................................................................................... 43 3.11 E{T)/E{1) versus T on a semi-log scale. (+): e = 0.45,& = 20,dn = 2,dg = 2. (o); Single logistic map (e = 0.0, a — 2.0) using k = 20, dn = 4,dg = 1. The lines depict: (A) scaling (3.10) with h = 0.7,