The University of Montana
Department of Mathematical Sciences

Technical report #8/2014

Generalized Correlation Integral Vectors: A New Distance Concept for Chaotic Dynamical Systems

Heikki Haario, Leonid Kalachev and Janne Hakkarainen

Abstract

Several concepts of dimension have been developed to characterize properties of chaotic trajectories. To estimate parameters of chaotic dynamical systems a measure to quantify the likelihood function of chaotic variability (the 'distance' between di?erent trajectories) is needed. We review problems encountered by previously used method and propose a method related to the correlation dimension concept. The major advantage of the new construct is its insensitivity with respect to varying initial values, to the choice of a solver, numeric tolerances, etc. A way to create the statistical likelihood for model parameters is presented, together with a sound framework for Markov chain Monte Carlo sampling. The methodology is illustrated using computational examples for the Lorenz 63 and Lorenz 95 systems.

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