### Abstract

We present a closed form stability criterion for the periodic orbits of two-dimensional conservative as well as "dissipative" mappings which are analogous to the Poincaré maps of dynamical systems. Our stability criterion has a particularly simple form involving a finite, symmetric, nearly tridiagonal determinant. Its derivation is based on an extension of the stability analysis of Hill's differential equation to difference equations. We apply our criterion and derive a sufficient stability condition for a large class of periodic orbits of the widely studied "standard mapping" describing a periodically "kicked" free rotator. As another example we also obtain explicitly and in closed form the intervals of bounded (and unbounded) solutions of a discrete "Schrödinger equation" for the Kronig and Penney crystal model.

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

Pages (from-to) | 1867-1877 |

Number of pages | 11 |

Journal | Journal of Mathematical Physics |

Volume | 22 |

Issue number | 9 |

Publication status | Published - 1981 |

Externally published | Yes |

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### ASJC Scopus subject areas

- Organic Chemistry

### Cite this

*Journal of Mathematical Physics*,

*22*(9), 1867-1877.

**On the stability of periodic orbits of two-dimensional mappings.** / Bountis, Tassos; Helleman, Robert H G.

Research output: Contribution to journal › Article

*Journal of Mathematical Physics*, vol. 22, no. 9, pp. 1867-1877.

}

TY - JOUR

T1 - On the stability of periodic orbits of two-dimensional mappings

AU - Bountis, Tassos

AU - Helleman, Robert H G

PY - 1981

Y1 - 1981

N2 - We present a closed form stability criterion for the periodic orbits of two-dimensional conservative as well as "dissipative" mappings which are analogous to the Poincaré maps of dynamical systems. Our stability criterion has a particularly simple form involving a finite, symmetric, nearly tridiagonal determinant. Its derivation is based on an extension of the stability analysis of Hill's differential equation to difference equations. We apply our criterion and derive a sufficient stability condition for a large class of periodic orbits of the widely studied "standard mapping" describing a periodically "kicked" free rotator. As another example we also obtain explicitly and in closed form the intervals of bounded (and unbounded) solutions of a discrete "Schrödinger equation" for the Kronig and Penney crystal model.

AB - We present a closed form stability criterion for the periodic orbits of two-dimensional conservative as well as "dissipative" mappings which are analogous to the Poincaré maps of dynamical systems. Our stability criterion has a particularly simple form involving a finite, symmetric, nearly tridiagonal determinant. Its derivation is based on an extension of the stability analysis of Hill's differential equation to difference equations. We apply our criterion and derive a sufficient stability condition for a large class of periodic orbits of the widely studied "standard mapping" describing a periodically "kicked" free rotator. As another example we also obtain explicitly and in closed form the intervals of bounded (and unbounded) solutions of a discrete "Schrödinger equation" for the Kronig and Penney crystal model.

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UR - http://www.scopus.com/inward/citedby.url?scp=0000778169&partnerID=8YFLogxK

M3 - Article

VL - 22

SP - 1867

EP - 1877

JO - Journal of Mathematical Physics

JF - Journal of Mathematical Physics

SN - 0022-2488

IS - 9

ER -