Global stability and bifurcations of perturbed Gumowski–Mira difference equation

Abstract

A generalized convergence theorem for higher order difference equations is established by quasi-Lyapunov function method. From this stability result we deduce the existence of global asymptotically stable fixed point and attractive two-periodic solution of the perturbed Gumowski–Mira difference equation. We also study global bifurcations of this system as the parameters vary. For instance we show that as the recombination coefficient moves through a critical curve, a fixed point loses its asymptotic stability and an attractive cycle of period 2 emerges near the fixed point due to a period-doubling bifurcation. The associated existence regions are also located.

Keywords::

• perturbation
• difference equation
• Lyapunov function
• asymptotic stability
• periodic solution

MSC (2010) Classification::

• 39A11
• 39A20

Acknowledgements

We thank the anonymous referees for their careful reading of the manuscript and their helpful comments.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work is supported by National Natural Science Foundation of China [grant number 11101283], [grant number 11202192], [grant number 11371140], [grant number 11178014]; and China Scholarship Council.