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Title: Generalized correlation integral vectors: A distance concept for chaotic dynamical systems

Several concepts of fractal dimension have been developed to characterise properties of attractors of chaotic dynamical systems. Numerical approximations of them must be calculated by finite samples of simulated trajectories. In principle, the quantities should not depend on the choice of the trajectory, as long as it provides properly distributed samples of the underlying attractor. In practice, however, the trajectories are sensitive with respect to varying initial values, small changes of the model parameters, to the choice of a solver, numeric tolerances, etc. The purpose of this paper is to present a statistically sound approach to quantify this variability. We modify the concept of correlation integral to produce a vector that summarises the variability at all selected scales. The distribution of this stochastic vector can be estimated, and it provides a statistical distance concept between trajectories. Here, we demonstrate the use of the distance for the purpose of estimating model parameters of a chaotic dynamic model. The methodology is illustrated using computational examples for the Lorenz 63 and Lorenz 95 systems, together with a framework for Markov chain Monte Carlo sampling to produce posterior distributions of model parameters.
Authors:
 [1] ;  [2] ;  [3]
  1. School of Engineering Science, Lappeenranta University of Technology, Lappeenranta (Finland)
  2. Department of Mathematical Sciences, University of Montana, Missoula, Montana 59812-0864 (United States)
  3. Earth Observation Unit, Finnish Meteorological Institute, Helsinki (Finland)
Publication Date:
OSTI Identifier:
22402562
Resource Type:
Journal Article
Resource Relation:
Journal Name: Chaos (Woodbury, N. Y.); Journal Volume: 25; Journal Issue: 6; Other Information: (c) 2015 Author(s); Country of input: International Atomic Energy Agency (IAEA)
Country of Publication:
United States
Language:
English
Subject:
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; APPROXIMATIONS; ATTRACTORS; CHAOS THEORY; CORRELATIONS; DISTANCE; DYNAMICS; FRACTALS; INTEGRALS; MARKOV PROCESS; MONTE CARLO METHOD; SAMPLING; SIMULATION; TOLERANCE; TRAJECTORIES; VECTORS