Computing dynamical curlicues

Description

A curlicue is a piece-wise linear curve in the complex plane which can be generated by an arbitrary sequence of real numbers u_n. It can be interpreted as a trajectory of a particle in the plane which starts in the origin at time t=0 and moves with a constant velocity, changing its direction at instances t=0,1,2,3,..., where the new direction is given by a number 2*pi u_t mod 2*pi. In particular, when the sequence u_n is given by iterates of some dynamical system (e.g. a circle homeomorphism) at a given point we speak about dynamically generated curlicues. This dataset contains source codes of the Matlab functions Rotation.m, Arnold.m and Sequence.m which can be used to plot the first N points of a curlicue generated, respectively, by rotation on the circle by arbitary angle, the Arnold circle map (with different parameters) and the u_n=nlog(n). Additionally, these functions allow us to calculate other properties of a curlicue, such as corresponding Birkhoff average and diameter of a curlicue. Description of the functions and variables involved is provided as comments in m-files. It's worth pointing out that the function Sequence.m can be easily modified to compute and draw a curlicue generated by an arbitrary sequence u_n given by explicit formula. We also include txt-files with exemplary data obtained by these functions and four figures (eps-files) generated by a function Sequence.m for various sequences. More details and precise definition of a curlicue is available in the file Curlicues_description, contained in the dataset and references therein

Dataset file

Curlicues_dane.zip

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Year of publication:

2020

Verification date:

2020-12-17

Dataset language:

English

Fields of science:

mathematics (Natural sciences)

DOI:

10.34808/anjg-q802

Verified by:

Gdańsk University of Technology

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