Analysis and anti-control of period-doubling bifurcation for the one-dimensional discrete system with three parameters and a square term

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.