### Abstract

We investigate why discretized versions f_{N} of one-dimensional ergodic maps f : I → I behave in many ways similarly to their continuous counterparts. We propose to register observations of the N × N discretization f_{N} on a coarse M × M grid, with N = cM, c being an integer. We prove that rounding errors behave like uniformly distributed random variables, and by assuming their independence, the M × M incidence matrix A^{M} associated with the continuous map (indicating which of the M equal subintervals is mapped onto which) can be expected to be identical to the incidence matrix B^{N,M} associated with the aforementioned coarse grid, if c ≥ √deg(f)^{N}, where deg(f) denotes the degree of f. We show how coarse-grained registration can be used as a "digital" definition of an unstable orbit and how this can be applied in real computations. Combination of these results with ideas from the random map model suggests an intuitive explanation for the statistical similarity between f and f_{N}. Our approach is not a rigorous one, however, we hope that the results will be useful for the computational community and may facilitate a rigourous mathematical description.

Original language | English |
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Pages (from-to) | 861-870 |

Number of pages | 10 |

Journal | International Journal of Bifurcation and Chaos in Applied Sciences and Engineering |

Volume | 15 |

Issue number | 3 |

DOIs | |

Publication status | Published - Mar 2005 |

### Fingerprint

### Keywords

- Chaos
- Coarse-grained model
- Discretization

### ASJC Scopus subject areas

- General
- Applied Mathematics

### Cite this

**Coarse-grained observation of discretized maps.** / Domokos, G.

Research output: Contribution to journal › Article

}

TY - JOUR

T1 - Coarse-grained observation of discretized maps

AU - Domokos, G.

PY - 2005/3

Y1 - 2005/3

N2 - We investigate why discretized versions fN of one-dimensional ergodic maps f : I → I behave in many ways similarly to their continuous counterparts. We propose to register observations of the N × N discretization fN on a coarse M × M grid, with N = cM, c being an integer. We prove that rounding errors behave like uniformly distributed random variables, and by assuming their independence, the M × M incidence matrix AM associated with the continuous map (indicating which of the M equal subintervals is mapped onto which) can be expected to be identical to the incidence matrix BN,M associated with the aforementioned coarse grid, if c ≥ √deg(f)N, where deg(f) denotes the degree of f. We show how coarse-grained registration can be used as a "digital" definition of an unstable orbit and how this can be applied in real computations. Combination of these results with ideas from the random map model suggests an intuitive explanation for the statistical similarity between f and fN. Our approach is not a rigorous one, however, we hope that the results will be useful for the computational community and may facilitate a rigourous mathematical description.

AB - We investigate why discretized versions fN of one-dimensional ergodic maps f : I → I behave in many ways similarly to their continuous counterparts. We propose to register observations of the N × N discretization fN on a coarse M × M grid, with N = cM, c being an integer. We prove that rounding errors behave like uniformly distributed random variables, and by assuming their independence, the M × M incidence matrix AM associated with the continuous map (indicating which of the M equal subintervals is mapped onto which) can be expected to be identical to the incidence matrix BN,M associated with the aforementioned coarse grid, if c ≥ √deg(f)N, where deg(f) denotes the degree of f. We show how coarse-grained registration can be used as a "digital" definition of an unstable orbit and how this can be applied in real computations. Combination of these results with ideas from the random map model suggests an intuitive explanation for the statistical similarity between f and fN. Our approach is not a rigorous one, however, we hope that the results will be useful for the computational community and may facilitate a rigourous mathematical description.

KW - Chaos

KW - Coarse-grained model

KW - Discretization

UR - http://www.scopus.com/inward/record.url?scp=21144442205&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=21144442205&partnerID=8YFLogxK

U2 - 10.1142/S021812740501248X

DO - 10.1142/S021812740501248X

M3 - Article

VL - 15

SP - 861

EP - 870

JO - International Journal of Bifurcation and Chaos in Applied Sciences and Engineering

JF - International Journal of Bifurcation and Chaos in Applied Sciences and Engineering

SN - 0218-1274

IS - 3

ER -