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PHASE-SPACE RECONSTRUCTION FOR SYMMETRICAL DYNAMIC-SYSTEMS
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UNSPECIFIED (1992) PHASE-SPACE RECONSTRUCTION FOR SYMMETRICAL DYNAMIC-SYSTEMS. PHYSICA D, 58 (1-4). pp. 216-228. ISSN 0167-2789.
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Abstract
Since the pioneering work of Packard et al. and Takens it has become customary to reconstruct the topology of attractors in phase space from a time series of one-dimensional experimental observations by using delay coordinates. Many practical refinements of the original methods have been developed.
Many experimental systems possess symmetry, and bifurcations can cause changes in the symmetry of observed states. These changes are quite subtle when the dynamics is chaotic. It is therefore important to reconstruct not just the topology of the attractor, but its symmetry. We indicate how this can be done by extending the Packard-Takens approach to a single equivariant observation, taking values not in the real numbers R but in a linear representation V of the symmetry group G. In effect a single set of symmetrically related observations is required. Our central point is that not all plausible choices for such a set can generate embeddings. In order for the method to produce an embedding, it is necessary that V should be "sufficiently complicated". More precisely, the phase space M must be subordinate to V in a sense introduced by Wassermann. This concept is technical, but unavoidable in this context, and it greatly clarifies the issue of embeddability. Using it, we state a symmetric version of the Takens embedding theorem, and sketch the proof. We also discuss the issue of "setwise" versus 'pointwise' symmetry of an attractor, and relate this to the transition from spatial order to spatial disorder in temporally chaotic systems.
Item Type: | Journal Article | ||||
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Subjects: | Q Science > QA Mathematics Q Science > QC Physics |
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Journal or Publication Title: | PHYSICA D | ||||
Publisher: | ELSEVIER SCIENCE BV | ||||
ISSN: | 0167-2789 | ||||
Official Date: | 15 September 1992 | ||||
Dates: |
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Volume: | 58 | ||||
Number: | 1-4 | ||||
Number of Pages: | 13 | ||||
Page Range: | pp. 216-228 | ||||
Publication Status: | Published |
Data sourced from Thomson Reuters' Web of Knowledge
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