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UNFOLDING THE TORUS - OSCILLATOR GEOMETRY FROM TIME DELAYS
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UNSPECIFIED (1993) UNFOLDING THE TORUS - OSCILLATOR GEOMETRY FROM TIME DELAYS. JOURNAL OF NONLINEAR SCIENCE, 3 (4). pp. 459-475. ISSN 0938-8974.
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Abstract
We present a simple method of plotting the trajectories of systems of weakly coupled oscillators. Our algorithm uses the time delays between the ''firings'' of the oscillators. For any system of n weakly coupled oscillators there is an attracting invariant n-dimensional torus, and the attractor is a subset of this invariant torus. The invariant torus intersects a suitable codimension-1 surface of section at an (n - 1)-dimensional torus. The dynamics of n coupled oscillators can thus be reduced, in principle, to the study of Poincare maps of the (n - 1)-dimensional torus. This paper gives a practical algorithm for measuring the n - 1 angles on the torus. Since visualization of 3 (or higher) dimensional data is difficult we concentrate on n = 3 oscillators. For three oscillators, a standard projection of the Poincare map onto the plane yields a projection of the 2-torus which is 4-to-1 over most of the torus, making it difficult to observe the structure of the attractor. Our algorithm allows a direct measurement of the 2 angles on the torus, so we can plot a 1-to-1 map from the invariant torus to the ''unfolded torus'' where opposite edges of a square are identified. In the cases where the attractor is a torus knot, the knot type of the attractor is obvious in our projection.
| Item Type: | Journal Article | ||||
|---|---|---|---|---|---|
| Subjects: | Q Science > QA Mathematics T Technology > TJ Mechanical engineering and machinery Q Science > QC Physics |
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| Journal or Publication Title: | JOURNAL OF NONLINEAR SCIENCE | ||||
| Publisher: | SPRINGER VERLAG | ||||
| ISSN: | 0938-8974 | ||||
| Official Date: | 1993 | ||||
| Dates: |
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| Volume: | 3 | ||||
| Number: | 4 | ||||
| Number of Pages: | 17 | ||||
| Page Range: | pp. 459-475 | ||||
| Publication Status: | Published |
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