Point Bifurcations and Bubbles for a Cubic Family

Abstract

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) = x^{3}-2x + c has a point bifurcation at c = 1/\sqrt {3} and a bubble in its bifurcation diagram.

Keywords:

• Point bifurcation
• Bubble
• Cubic family
• Bifurcation theory
• 37E05
• 37G25
• 37C35

Acknowledgements

The author would like to thank Dr. Bau-Sen Du for introducing the interesting problems related to the family f_{c}(x) = x^{3}-2x + c.