## Abstract

A rotation number in the case of one-dimensional maps is introduced. As is shown, this rotation number is equivalent to the already known rotation number in the case of two-dimensional maps. The definition of the rotation number is given in two steps. First, it is defined for periodic orbits inside a window of organized motion (WOM). We show that in this case our definition coincides with the definition of the over-rotation number. Then, our definition is further generalized for chaotic orbits outside the WOMs. Thus, we obtain a unified definition of the rotation number for the whole area of the chaotic zone of the bifurcation diagram, having a number of useful applications. Namely, it can be used as a tool to distinguish whether an orbit is contained within a WOM or not, as a tool of numerical location of the bifurcation points, of the band mergings, as well as of the boundary points of a WOM. Finally, a method of numerical calculation of the percentage of the cumulative width of the WOMs in every particular segment (chaotic band) of the chaotic zone is given.

Original language | English (US) |
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Article number | 011 |

Pages (from-to) | 15231-15244 |

Number of pages | 14 |

Journal | Journal of Physics A: Mathematical and General |

Volume | 39 |

Issue number | 49 |

DOIs | |

State | Published - Dec 8 2006 |

Externally published | Yes |

## All Science Journal Classification (ASJC) codes

- Statistical and Nonlinear Physics
- Mathematical Physics
- General Physics and Astronomy