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On Euler-like discrete models of the logistic differential equation

Authors
  • Grote, K.
  • Meyer-Spasche, R.
  • Max-Planck-Institut fuer Plasmaphysik, G...
Publication Date
Jan 01, 1995
Source
OpenGrey Repository
Keywords
Language
English
License
Unknown

Abstract

The stability theory of difference schemes is mostly a linear theory. To understand the behavior of difference schemes on nonlinear differential equations, it seems desirable to extend the stability theory into a nonlinear theory. As a step in that direction, we investigate the stability properties of Euler-related integration algorithms by checking how they preserve and violate the dynamical structure of the logistic differential equation. We find that partially implicit rational schemes are superior to explicit schemes when they are stable and the blow-up time has not passed. When such a rational scheme turns unstable, however, it has much less desirable properties than explicit schemes. As a side product of these investigations, we found a map with two branches of stable fixed points. Both of them lose stability to a Feigenbaum sequence of period doubling bifurcations and chaotic trajectories independently of each other. To our knowledge, this is the first such example. (orig.) / 19 refs. / Available from TIB Hannover: RA 71(6/330) / FIZ - Fachinformationszzentrum Karlsruhe / TIB - Technische Informationsbibliothek / SIGLE / DE / Germany

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