Point bifurcations and bubbles for a cubic family

Ming-Chia Li*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

5 Scopus citations


In this paper, we give a sufficient condition for point bifurcations. We also study bubbles in bifurcations which indicates a transition from simple dynamics to the chaotic one and then back to simple ones. We prove that the cubic family f(c, x) = x3 - 2x + c has a point bifurcation at c = 1/√3 and a bubble in its bifurcation diagram.

Original languageEnglish
Pages (from-to)553-558
Number of pages6
JournalJournal of Difference Equations and Applications
Issue number6
StatePublished - 1 Jun 2003


  • Bifurcation theory
  • Bubble
  • Cubic family
  • Point bifurcation

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