### Abstract

The accurate computation of periodic orbits and the precise knowledge of their bifurcation properties are very important for studying the behavior of many dynamical systems of physical interest. In this paper, we present an iterative method for computing periodic orbits, which has the advantage of improving the convergence of previous Newton-like schemes, especially near bifurcation points. This method is illustrated here on a conservative, nonlinear Mathieu equation, for which a sequence of period-doubling bifurcations is followed, long enough to obtain accurate estimates of the two universal scaling constants α, β, as well as the universal rate δ, by which the bifurcation values of a parameter q = qk, k =1, 2, 3, ..., tend to their limiting value, q_{∞} < ∞, as k increases.

Original language | English |
---|---|

Pages (from-to) | 1-14 |

Number of pages | 14 |

Journal | Journal of Computational Physics |

Volume | 88 |

Issue number | 1 |

DOIs | |

Publication status | Published - 1990 |

Externally published | Yes |

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### ASJC Scopus subject areas

- Computer Science Applications
- Physics and Astronomy(all)

### Cite this

*Journal of Computational Physics*,

*88*(1), 1-14. https://doi.org/10.1016/0021-9991(90)90239-W

**A convergence-improving iterative method for computing periodic orbits near Bifurcation Points.** / Vrahatis, Michael N.; Bountis, Tassos; Budinsky, Nurit.

Research output: Contribution to journal › Article

*Journal of Computational Physics*, vol. 88, no. 1, pp. 1-14. https://doi.org/10.1016/0021-9991(90)90239-W

}

TY - JOUR

T1 - A convergence-improving iterative method for computing periodic orbits near Bifurcation Points

AU - Vrahatis, Michael N.

AU - Bountis, Tassos

AU - Budinsky, Nurit

PY - 1990

Y1 - 1990

N2 - The accurate computation of periodic orbits and the precise knowledge of their bifurcation properties are very important for studying the behavior of many dynamical systems of physical interest. In this paper, we present an iterative method for computing periodic orbits, which has the advantage of improving the convergence of previous Newton-like schemes, especially near bifurcation points. This method is illustrated here on a conservative, nonlinear Mathieu equation, for which a sequence of period-doubling bifurcations is followed, long enough to obtain accurate estimates of the two universal scaling constants α, β, as well as the universal rate δ, by which the bifurcation values of a parameter q = qk, k =1, 2, 3, ..., tend to their limiting value, q∞ < ∞, as k increases.

AB - The accurate computation of periodic orbits and the precise knowledge of their bifurcation properties are very important for studying the behavior of many dynamical systems of physical interest. In this paper, we present an iterative method for computing periodic orbits, which has the advantage of improving the convergence of previous Newton-like schemes, especially near bifurcation points. This method is illustrated here on a conservative, nonlinear Mathieu equation, for which a sequence of period-doubling bifurcations is followed, long enough to obtain accurate estimates of the two universal scaling constants α, β, as well as the universal rate δ, by which the bifurcation values of a parameter q = qk, k =1, 2, 3, ..., tend to their limiting value, q∞ < ∞, as k increases.

UR - http://www.scopus.com/inward/record.url?scp=0011332685&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0011332685&partnerID=8YFLogxK

U2 - 10.1016/0021-9991(90)90239-W

DO - 10.1016/0021-9991(90)90239-W

M3 - Article

VL - 88

SP - 1

EP - 14

JO - Journal of Computational Physics

JF - Journal of Computational Physics

SN - 0021-9991

IS - 1

ER -