Analysis and anti-control of bifurcations in two-dimensional, three-parameter discrete dynamical system with cubic terms

In this paper, we proposed a class of two-dimensional three-parameter discrete dynamical systems with cubic terms, which can be applied to image encryption. We present a study on the analysis and control of the bifurcations in these systems. Initially, the existence and stability conditions of the fixed points of the proposed systems were proposed. Subsequently, based on the center manifold and bifurcation theories, we determined the conditions for the existence of Neimark-Sacker, pitchfork, and period-doubling bifurcations. The bifurcation diagram and the phase portraits were employed in the numerical experiments to verify the correctness of theoretical analysis. Finally, anti-controllers were used to induce Neimark-Sacker and period-doubling bifurcations, which were designed by integrating the bifurcation conditions with the state feedback method. The proposed anti-controllers caused the systems to undergo the desired bifurcations at the preset parameter values. Numerical simulations verified the effectiveness and robustness of the proposed controllers.