Verifying chaotic dynamics from experimental data

Abstract

ySchool of Mathematics and Statistics, University of Western Australia, Crawley, WA 6009, AustraliazDepartment of Mathematics and Statistics, University of Melbourne, Parkvillle, Victoria 3052, AustraliaEmail: michael.small@uwa.edu.auAbstract—Often, one is faced with measured time se-ries data from some (presumed to be deterministic) dynam-ical system. The problem is to correctly infer the true, or atleast, likely, underlying dynamical system from data alone.A variety of methods exist to achieve this — under the gen-eral umbrella on nonlinear modeling and machine learning.These methods fit a surface (usually smooth) to the data insuch a way that that surface can be used as a proxy for theevolution operator of the original system. Unfortunately,di erent methods produce di erent results. Worse still, dueto the nonlinearity inherent to the problem, even the samemethod will produce a range of distinct local minima. Theaim of this report is to apply an ensemble of dynamicalmeasures of system behaviour to show how one can deter-mine which models behave most like the underlying data.1. IntroductionFor most experimental nonlinear systems it is not possi-ble to write down a closed-form analytic description of thedynamics. In such situations it is therefore appealing to ap-ply the methods of delay reconstruction [7] and nonlinearmodelling [4] to estimate the underlying evolution opera-tor. One can then use this estimate to study properties of thedynamical system which it represents, and then extrapolatethis to the experimental system of interest. This secondstep can be rather problematic. In 2009, Small and Carmeli[5] re-examined an earlier work of Marquet and co-workers[3]. Marquet et al. [3] sought to construct a global nonlin-ear model from time series data of Canadian Lynx popu-lations. Based on their models they demonstrated that thedata is potentially consistent with chaos — moreover, theywere able to claim that the chaotic dynamics of the modelswas direct evidence of chaos in a real ecosystem. Whileit is true that their model — a best fit to the observed data— did indeed exhibit the desired “interesting” chaotic dy-namics, it would not be true to infer that this is the onlyreasonable explanation of this data. In [5] we showed thatit is equally likely that the data could be described as a pe-riodic orbit. Moreover, for our models (using a di6 erentmodeling algorithm from Marquet and co-workers, whichwe will explain below) when interesting chaotic dynamicsdid arise this turned out to be transient — with a time scalemuch longer than the time scale of the original data. In Fig.1 we plot the original data used by both [3] and [5] togetherwith an ensemble of these model simulations.