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Detection of all low-period windows for the logistic map | IEEE Conference Publication | IEEE Xplore

Detection of all low-period windows for the logistic map


Abstract:

A systematic method to find all low-period windows for the logistic map fa(x) = ax(1 - x) is proposed. The method is used to obtain very good approximations of positions ...Show More

Abstract:

A systematic method to find all low-period windows for the logistic map fa(x) = ax(1 - x) is proposed. The method is used to obtain very good approximations of positions of periodic windows with periods p ≤ 20. For each window using the forward shooting based interval Newton operator we confirm its existence by proving the existence of a sink for several parameter values inside the window and compute a very accurate rigorous lower bound of the window width.
Date of Conference: 24-27 May 2015
Date Added to IEEE Xplore: 30 July 2015
Electronic ISBN:978-1-4799-8391-9

ISSN Information:

Conference Location: Lisbon, Portugal

I. Introduction

The logistic map (frequently referred to as the quadratic map) popularised in [1] is perhaps the most well known example of a nonlinear dynamical system that can exhibit complex behaviour. In spite of many achievements in understanding the dynamics of this system some problems are still open. One of them is the measure of sets of parameter values for which the map is regular (periodic) or chaotic. It is known that these two sets are disjoint, their measures are positive [2], and that almost every parameter produces either periodic or chaotic behavior, i.e. the union of this two sets has full measure [3]. One possible approach to obtain bounds for measures of these two sets is based on finding all short periodic windows. Periodic windows can be found analytically only for very low periods. The bisection method to find periodic windows for the logistic map has been used in [4]. In this method the interval of parameter values is split into smaller parts, the Newton method is used to find superstable orbits in each part and then the regions found are extended to cover as much of a periodic window as possible. In [4], the results on widths of periodic windows found were used to obtain a lower bound on the measure of the set of stable parameters.

References

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