Technical Brief

On a Comprehensive Topological Analysis of Moore Spiegel Attractor

[+] Author and Article Information
Anirban Ray

Department of Physics,
Gour Mahavidalya,
Malda 732142, West Bengal, India
e-mail: anirban.chaos@gmail.com

A. RoyChowdhury

High Energy Physics Division,
Department of Physics,
Jadavpur University,
Kolkata 700 032, India
e-mail: arc.roy@gmail.com

1Corresponding author.

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received June 18, 2016; final manuscript received July 3, 2017; published online October 9, 2017. Assoc. Editor: Hiroshi Yabuno.

J. Comput. Nonlinear Dynam 13(1), 014501 (Oct 09, 2017) (7 pages) Paper No: CND-16-1288; doi: 10.1115/1.4037413 History: Received June 18, 2016; Revised July 03, 2017

A topological analysis of the attractor associated with the Moore–Spiegel nonlinear system is performed, following the basic idea laid down by Gilmore and Lefranc (2002, The Topology of Chaos, Wiley, Hoboken, NJ). Starting with the usual fixed point analysis and their stability, we proceed to study in detail the process of chaotic orbit extraction with the help of close return map. This is then used to construct the symbolic dynamics associated with it, which is helpful in understanding the sequential change taking place inside the attractor. In the next part, we show how to characterize the evolution of the attractor from its birth to the crisis by finding out the homoclinic orbit and the corresponding unstable manifold. In the concluding part of the paper, we show how all the pertinent information of the attractor can be encoded in the template, leading to the explicit realization of linking numbers and the relative rotation rates. In the concluding section, we have touched upon a new approach to chaotic dynamics, using the flow curvature manifold to display the relative positioning of the attractor in relation to the fixed points and the null lines.

Copyright © 2018 by ASME
Topics: Attractors
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Grahic Jump Location
Fig. 1

Various phase space structure at R = 110 and T = 42.9

Grahic Jump Location
Fig. 2

(a) Histogram of close return plot at T = 42.9 and R = 110.0 and (b) histogram of close return plot at T = 38.0 and R = 100.0

Grahic Jump Location
Fig. 3

UPO extracted from the attractor at R = 110 and T = 42.9

Grahic Jump Location
Fig. 4

UPO extracted from the attractor at R = 100 and T = 38

Grahic Jump Location
Fig. 5

Poincare section at R = 110 and T = 42.9

Grahic Jump Location
Fig. 6

Bifurcation of Moore–Spiegel system with the variation of R when T = 42.9

Grahic Jump Location
Fig. 8

Ribbon-like formation of attractor at R = 160.6 and T = 42.9

Grahic Jump Location
Fig. 7

(a) Magnification of portion of bifurcation near Rc = 120.66, (b) xy projection of attractor at Rc = 118, and (c) xy projection of attractor at Rc = 126

Grahic Jump Location
Fig. 9

Template structure of Moore–Spiegel attractor. Here, flow is in anti-clockwise direction. Direction of flow is denoted through an arrow.

Grahic Jump Location
Fig. 10

Attractor with unstable manifold

Grahic Jump Location
Fig. 11

Attractor with unstable manifold and null lines manifold



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