Bifurcations and complex instability in a 4-dimensional symplectic mapping View Full Text


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Article Info

DATE

1988-03

AUTHORS

G. Contopoulos, A. Giorgilli

ABSTRACT

We study the main periodic solutions of a 4-dimensional symplectic mapping composed of two coupled 2-dimensional mappings. Their bifurcations were calculated numerically and also theoretically for small values of the coupling parameter μ. Most bifurcating families of period 2n (n≻0) have complex unstable regions that extend from μ=0 to the maximum allowed value of μ for each family. These complex unstable regions do not allow the transmisssion of the stability of the corresponding families to families of higher order. We found only one family with a complex unstable region not extending to the maximum μ, but in this case also the transmission of the stability is stopped at an inverse bifurcation. Thus although there are infinite sequences of bifurcations (of the Feigenbaum type) in the limiting 2-dimensional case μ=0, all such sequences are interrupted when the system is 4-dimensional (i.e. for μ≠0). The appearance of complex instability for μ=0 can be predicted by studying the cases μ=0 and applying the Krein-Moser theorem. More... »

PAGES

19-28

References to SciGraph publications

  • 1980-05. Universal properties in conservative dynamical systems in LETTERE AL NUOVO CIMENTO (1971-1985)
  • 1976-06. Simple mathematical models with very complicated dynamics in NATURE
  • 1983-10. Termination of sequences of bifurcation in 3-dimensional Hamiltonian systems in LETTERE AL NUOVO CIMENTO (1971-1985)
  • 1899-07. Les méthodes nouvelles de la mécanique céleste in IL NUOVO CIMENTO (1895-1900)
  • Journal

    TITLE

    Meccanica

    ISSUE

    1

    VOLUME

    23

    Author Affiliations

    Identifiers

    URI

    http://scigraph.springernature.com/pub.10.1007/bf01561006

    DOI

    http://dx.doi.org/10.1007/bf01561006

    DIMENSIONS

    https://app.dimensions.ai/details/publication/pub.1004183177


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