Volume 47, Number 6, September II 1999
|Page(s)||649 - 655|
|Published online||01 September 2002|
Nonlinear dependencies between sets of periodic orbits
Departamento de Física, Faculdade de Ciências, Universidade de Lisboa 1749-016 Lisboa, Portugal Instituto de Física, Universidade Federal do Rio Grande do Sul 91501-970 Porto Alegre, Brazil
Accepted: 19 July 1999
We report the existence of a phenomenon of inheritance among periodic trajectories of the quadratic map. Within the set of all possible trajectories of period k we find "mother” trajectories from which several "daughters” may be derived by simple polynomial transformations. For example, from the six orbital points zi of a period-six mother we get three additional period-six daughters as the zeros of the six cubics . Daughters might have daughters. This stratification shows that periodic orbits are not necessarily independent of each other. This fact could be of importance for decomposing certain sums involving sets of periodic trajectories, particularly for trace formulas underlying semiclassical interpretations of spectra in atomic physics.
PACS: 05.45.-a – Nonlinear dynamics and nonlinear dynamical systems / 45.05.+x – General theory of classical mechanics of discrete systems / 03.65.Fd – Algebraic methods
© EDP Sciences, 1999
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