### 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 Poincaré 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 Poincaré 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.

Original language | English (US) |
---|---|

Pages (from-to) | 459-475 |

Number of pages | 17 |

Journal | Journal of Nonlinear Science |

Volume | 3 |

Issue number | 1 |

DOIs | |

State | Published - Dec 1993 |

### Fingerprint

### Keywords

- AMS numbers: 34C, 58F
- coupled oscillators
- data visualization
- torus maps

### ASJC Scopus subject areas

- Statistical and Nonlinear Physics
- Mathematical Physics
- Applied Mathematics
- Mathematics(all)
- Mechanics of Materials
- Computational Mechanics

### Cite this

*Journal of Nonlinear Science*,

*3*(1), 459-475. https://doi.org/10.1007/BF02429874

**Unfolding the torus : Oscillator geometry from time delays.** / Ashwin, P.; Swift, James W.

Research output: Contribution to journal › Article

*Journal of Nonlinear Science*, vol. 3, no. 1, pp. 459-475. https://doi.org/10.1007/BF02429874

}

TY - JOUR

T1 - Unfolding the torus

T2 - Oscillator geometry from time delays

AU - Ashwin, P.

AU - Swift, James W

PY - 1993/12

Y1 - 1993/12

N2 - 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 Poincaré 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 Poincaré 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.

AB - 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 Poincaré 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 Poincaré 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.

KW - AMS numbers: 34C, 58F

KW - coupled oscillators

KW - data visualization

KW - torus maps

UR - http://www.scopus.com/inward/record.url?scp=51249169477&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=51249169477&partnerID=8YFLogxK

U2 - 10.1007/BF02429874

DO - 10.1007/BF02429874

M3 - Article

AN - SCOPUS:51249169477

VL - 3

SP - 459

EP - 475

JO - Journal of Nonlinear Science

JF - Journal of Nonlinear Science

SN - 0938-8974

IS - 1

ER -