On the stability of periodic orbits of two-dimensional mappings

Tassos Bountis, Robert H G Helleman

Research output: Contribution to journalArticle

28 Citations (Scopus)

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 languageEnglish
Pages (from-to)1867-1877
Number of pages11
JournalJournal of Mathematical Physics
Volume22
Issue number9
Publication statusPublished - 1981
Externally publishedYes

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Stability criteria
Periodic Orbits
Orbits
orbits
Stability Criteria
Convergence of numerical methods
Closed-form
Difference equations
Unbounded Solutions
Hill Equation
Dynamical systems
Differential equations
Tridiagonal matrix
Bounded Solutions
Discrete Equations
Stability Condition
Difference equation
Crystals
difference equations
Stability Analysis

ASJC Scopus subject areas

  • Organic Chemistry

Cite this

Bountis, T., & Helleman, R. H. G. (1981). On the stability of periodic orbits of two-dimensional mappings. Journal of Mathematical Physics, 22(9), 1867-1877.

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

In: Journal of Mathematical Physics, Vol. 22, No. 9, 1981, p. 1867-1877.

Research output: Contribution to journalArticle

Bountis, T & Helleman, RHG 1981, 'On the stability of periodic orbits of two-dimensional mappings', Journal of Mathematical Physics, vol. 22, no. 9, pp. 1867-1877.
Bountis, Tassos ; Helleman, Robert H G. / On the stability of periodic orbits of two-dimensional mappings. In: Journal of Mathematical Physics. 1981 ; Vol. 22, No. 9. pp. 1867-1877.
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