A topological invariant of the Lorenz attractor

Reshetikhin and Turaev [10] gave a method to construct a topological invariant of compact oriented 3-manifolds from a ribbon Hopf algebra (e.g. a quantum group Uq(sl2)) using finite-dimensional representations of it. In this paper we give another independent method to construct a topological invariant of compact oriented 3-manifolds from a ribbon Hopf algebra via universal invariants of framed links without using representations of the algebra. For Uq(sl2) these two methods give different invariants of 3-manifolds.

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A topological invariant for nonlinear rotations of

Nonlinearity ◽

10.1088/0951-7715/10/1/010 ◽

1997 ◽

Vol 10 (1) ◽

pp. 153-158 ◽

Cited By ~ 1

Author(s):

Elisabeth Pécou

Keyword(s):

Topological Invariant

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A Study On Methods for Determining Phase Space Reconstruction Parameters

Journal of Computational and Nonlinear Dynamics ◽

10.1115/1.4052721 ◽

2021 ◽

Author(s):

Shihui Lang ◽

Zhu Hua ◽

Guodong Sun ◽

Yu Jiang ◽

Chunling Wei

Keyword(s):

Time Delay ◽

Phase Space ◽

Correlation Dimension ◽

Nearest Neighbor ◽

Correlation Method ◽

Phase Space Reconstruction ◽

Lorenz Attractor ◽

Embedding Dimension ◽

Phase Trajectories ◽

Auto Correlation

Abstract Several pairs of algorithms were used to determine the phase space reconstruction parameters to analyze the dynamic characteristics of chaotic time series. The reconstructed phase trajectories were compared with the original phase trajectories of the Lorenz attractor, Rössler attractor, and Chens attractor to obtain the optimum method for determining the phase space reconstruction parameters with high precision and efficiency. The research results show that the false nearest neighbor method and the complex auto-correlation method provided the best results. The saturated embedding dimension method based on the saturated correlation dimension method is proposed to calculate the time delay. Different time delays are obtained by changing the embedding dimension parameters of the complex auto-correlation method. The optimum time delay occurs at the point where the time delay is stable. The validity of the method is verified through combing the application of correlation dimension, showing that the proposed method is suitable for the effective determination of the phase space reconstruction parameters.

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A TORSIONAL TOPOLOGICAL INVARIANT

International Journal of Modern Physics A ◽

10.1142/s0217751x07038414 ◽

2007 ◽

Vol 22 (29) ◽

pp. 5237-5244 ◽

Cited By ~ 55

Author(s):

H. T. NIEH

Keyword(s):

Affine Connection ◽

Euler Class ◽

Christoffel Symbol ◽

Space Time ◽

Topological Invariant ◽

Point Of View ◽

Underlying Space ◽

Recent Developments ◽

Theory Point ◽

Curvature And Torsion

Curvature and torsion are the two tensors characterizing a general Riemannian space–time. In Einstein's general theory of gravitation, with torsion postulated to vanish and the affine connection identified to the Christoffel symbol, only the curvature tensor plays the central role. For such a purely metric geometry, two well-known topological invariants, namely the Euler class and the Pontryagin class, are useful in characterizing the topological properties of the space–time. From a gauge theory point of view, and especially in the presence of spin, torsion naturally comes into play, and the underlying space–time is no longer purely metric. We describe a torsional topological invariant, discovered in 1982, that has now found increasing usefulness in recent developments.

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A new invariant and decompositions of manifolds

Journal of Knot Theory and Its Ramifications ◽

10.1142/s0218216518500190 ◽

2018 ◽

Vol 27 (02) ◽

pp. 1850019

Author(s):

Eiji Ogasa

Keyword(s):

Topological Invariant ◽

The Body ◽

Precise Definition ◽

Closed Manifold ◽

Compact Manifolds ◽

Connected Components ◽

Fixed Base

We introduce a new topological invariant [Formula: see text] of compact manifolds-with-boundaries [Formula: see text] which is much connected with boundary-unions. A boundary-union is a kind of decomposition of compact manifolds-with-boundaries. See the body of the paper for the precise definition. Let [Formula: see text] and [Formula: see text] be [Formula: see text]-dimensional compact manifolds-with-boundaries. Let [Formula: see text] be a boundary-union of [Formula: see text] and [Formula: see text]. Then we have [Formula: see text] We define [Formula: see text] as follows: First, define an invariant of [Formula: see text]-closed manifolds. Take the maximum of the invariant of all connected-components of the boundary of each handle-body of an ordered-handle-decomposition with a fixed base [Formula: see text], where we impose the condition that the base [Formula: see text] is a (not necessarily connected) closed manifold. Take the minimum of the maximum for all ordered-handle-decompositions with the base [Formula: see text]. It is our another invariant [Formula: see text]. Take the maximum of the minimum, [Formula: see text], for all basis to satisfy the above condition. It is [Formula: see text]. See the body of the paper for the precise definition.

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