Europhys. Lett, 47 (6), pp. 649-655 (1999)
Nonlinear dependencies between sets of periodic orbits
J. A. C. Gallas
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
(received 20 May 1999; accepted in final form 19 July 1999)
PACS. 05.45 - Nonlinear dynamics and nonlinear
PACS. 45.05 - General theory of classical mechanics of discrete systems.
PACS. 03.65Fd - Algebraic methods.
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 x3-3x-zi=0. 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.
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