### 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 |
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Pages (from-to) | 1867-1877 |

Number of pages | 11 |

Journal | Journal of Mathematical Physics |

Volume | 22 |

Issue number | 9 |

DOIs | |

Publication status | Published - Jan 1 1981 |

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### ASJC Scopus subject areas

- Statistical and Nonlinear Physics
- Mathematical Physics

### Cite this

*Journal of Mathematical Physics*,

*22*(9), 1867-1877. https://doi.org/10.1063/1.525159