Visualizing the Template of a Chaotic Attractor

Abstract

0000−0001−9926−058X] , Jeff Meder 1[0000−0002−2360−8487] , Emmanuel Kieffer 2[0000−0002−5530−8577] , Raphaël Bleuse 1[0000−0002−6728−2132] , Martin Rosalie 2[0000−0003−3676−120X] , Grégoire Danoy 1[0000−0001−9419−4210] , and Pascal Bouvry 1,2[0000−0001−9338−2834]Abstract. Chaotic attractors are solutions of deterministic processes, of which the topology can be described by templates. We consider templates of chaotic attractors bounded by a genus-1 torus described by a linking matrix. This article introduces a novel and unique tool to validate a linking matrix, to optimize the compactness of the corresponding template and to draw this template. The article provides a detailed description of the different validation steps and the extraction of an order of crossings from the linking matrix leading to a template of minimal height. Finally, the drawing process of the template corresponding to the matrix is saved in a Scalable Vector Graphics (SVG) file.

Computing Height-Optimal Tangles Faster

“…We give an exact algorithm for simple lists running in O(n!ϕ n ) time, where ϕ = √ 5+1 2 ≈ 1.618 is the golden ratio, and an exact algorithm for general lists running in O((2|L|/n 2 +1) n 2 /2 ϕ n n) time, which is polynomial in |L| for fixed n ≥ 2 (see Section 3). We implemented the algorithm for general lists and compared it to the algorithm of Olszewski et al [7] using their benchmark set (see Section 4). We show that the asymptotic runtimes of the algorithms of Olszewski et al [7] for simple and for general lists are O(ϕ 2|L| 5 −|L|/n n) and 2 O(n 2 ) , respectively.…”

“…Inspired by the practical research of Olszewski et al [7], we have considered tangle-height minimization. We have shown that the problem is NP-hard, but we note that membership in NP is not obvious because the minimum height can be exponential in the size of the input.…”

“…We implemented the algorithm for general lists and compared it to the algorithm of Olszewski et al [7] using their benchmark set (see Section 4). We show that the asymptotic runtimes of the algorithms of Olszewski et al [7] for simple and for general lists are O(ϕ 2|L| 5 −|L|/n n) and 2 O(n 2 ) , respectively. In Section 5, we discuss feasibility for lists with simple structure.…”

“…Later Mindlin et al [6] showed how to characterize attractors using integer matrices that contain numbers of swaps between the orbits. Our research is based on a recent paper of Olszewski et al [7]. In the framework of their paper, one is given a set of wires that hang off a horizontal line in a fixed order, and a multiset of swaps between the wires; a tangle then is a visualization of these swaps, i.e., an order in which the swaps are performed, where only adjacent wires can be swapped and disjoint swaps can be done simultaneously.…”

“…Framework, Terminology, and Notation. We modify the terminology of Olszewski et al [7] in order to introduce a formal algebraic framework for the problem. Given n wires, a (swap) list L = (l ij ) of order n is a symmetric n × n matrix with non-negative entries and zero diagonal.…”