@article{Liu2025, 
author = {Limei Liu and Xitong Zhong},
title = {Analysis and anti-control of period-doubling bifurcation for the one-dimensional discrete system with three parameters and a square term},
year = {2025},
journal = {AIMS Mathematics},
volume = {10},
number = {2},
pages = {3227-3250},
keywords = {anti-control, period-doubling bifurcation, stability analysis, one-dimensional discrete system with three parameters and a square term, chaos, image encryption},
url = {https://www.sciopen.com/article/10.3934/math.2025150},
doi = {10.3934/math.2025150},
abstract = {In certain nonlinear systems, period-doubling bifurcations are a common way to cause chaos. Additionally, bifurcation advance or delay can be realized using anti-control of period-doubling bifurcation. To address the practical needs of engineering, anti-control of period-doubling bifurcation is a typical method of applying chaos. Based on these reasons, we conducted the following research: First, we proposed a new one-dimensional discrete system with three parameters and a square term. Existence and stability at the fixed point were studied for the one-dimensional discrete system with three parameters and a square term. Furthermore, bifurcation theory was used to determine the conditions of existence for transcritical bifurcation and period-doubling bifurcation. Numerical experiments verified the theoretical assessments of the bifurcation's results. Then, the state linear feedback control approach was used to implement the anti-control of period-doubling bifurcation in order to realize period-doubling bifurcation advance for the one-dimensional discrete system with three parameters and a square term. The conditions of the appropriate control parameters were analyzed in theory. Numerical experiments confirmed the efficiency and robustness of anti-control of period-doubling bifurcation for the one-dimensional discrete system with three parameters and a square term. The one-dimensional discrete system with three parameters and a square term with the anti-controller has advantages in image encryption.}
}