Correlation integral and determinism for a family of $2^\infty$ maps

Jana  Majerová

doi:10.3934/dcds.2016020

Article Contents

2016, Volume 36, Issue 9: 5067-5096. Doi: 10.3934/dcds.2016020

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Correlation integral and determinism for a family of $2^\infty$ maps

Jana Majerová^1,

1.
Slovanet a.s., Záhradnícka 151, 821 08 Bratislava

Received: May 31, 2015

Revised: February 29, 2016

Published: May 2017

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Abstract

The correlation integral and determinism are quantitative characteristics of a dynamical system based on the recurrence of orbits. For strongly non-chaotic interval maps, the determinism equals $1$ for every small enough threshold. This means that trajectories of such systems are perfectly predictable in the infinite horizon. In this paper we study the correlation integral and determinism for the family of $2^\infty$ non-chaotic maps, first considered by Delahaye in 1980. The determinism in a finite horizon equals $1$. However, the behaviour of the determinism in the infinite horizon is counter-intuitive. Sharp bounds on the determinism are provided.

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Mathematics Subject Classification: Primary: 37E05, 37E15; Secondary: 37M25.

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