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Convergent Chaos

Pradas, Marc; Pumir, Alain; Huber, Greg and Wilkinson, Michael (2017). Convergent Chaos. Journal of Physics A: Mathematical and Theoretical, 50(27), article no. 275101.

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DOI (Digital Object Identifier) Link: https://doi.org/10.1088/1751-8121/aa734d
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Abstract

Chaos is widely understood as being a consequence of sensitive dependence upon initial conditions. This is the result of an instability in phase space, which separates trajectories exponentially. Here, we demonstrate that this criterion should be refined. Despite their overall intrinsic instability, trajectories may be very strongly convergent in phase space over extremely long periods, as revealed by our investigation of a simple chaotic system (a realistic model for small bodies in a turbulent flow). We establish that this strong convergence is a multi-facetted phenomenon, in which the clustering is intense, widespread and balanced by lacunarity of other regions. Power laws, indicative of scale-free features, characterize the distribution of particles in the system. We use large-deviation and extreme-value statistics to explain the effect. Our results show that the interpretation of the 'butterfly effect' needs to be carefully qualified. We argue that the combination of mixing and clustering processes makes our specific model relevant to understanding the evolution of simple organisms. Lastly, this notion of convergent chaos, which implies the existence of conditions for which uncertainties are unexpectedly small, may also be relevant to the valuation of insurance and futures contracts.

Item Type: Journal Item
Copyright Holders: 2017 IOP Publishing
ISSN: 1751-8113
Project Funding Details:
Funded Project NameProject IDFunding Body
Not SetPHY11-25915National Science Foundation
Keywords: chaos; Lyapunov exponents; extreme values
Academic Unit/School: Faculty of Science, Technology, Engineering and Mathematics (STEM) > Mathematics and Statistics
Faculty of Science, Technology, Engineering and Mathematics (STEM)
Item ID: 50172
Depositing User: Marc Pradas
Date Deposited: 11 Jul 2017 15:10
Last Modified: 24 May 2019 08:52
URI: http://oro.open.ac.uk/id/eprint/50172
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