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

Limei Liu; Jiangna Ruan; Jun Zhai; Xiuling Li; Limei Liu; Jiangna Ruan; Jun Zhai; Xiuling Li

doi:10.3934/math.2026050

AIMS Mathematics

2026, Volume 11, Issue 1: 1175-1201. doi: 10.3934/math.2026050

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Research article

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

School of Statistics and Data Science, Jilin University of Finance and Economics, Changchun 130117, China

Received: 30 October 2025 Revised: 04 January 2026 Accepted: 09 January 2026 Published: 15 January 2026
MSC : 39A28, 39A30, 39A33, 65P30, 93B52

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.
- two-dimensional discrete dynamical system,
- stability,
- Neimark-Sacker bifurcation,
- period-doubling bifurcation,
- pitchfork bifurcation,
- anti-control
Citation: Limei Liu, Jiangna Ruan, Jun Zhai, Xiuling Li. Analysis and anti-control of bifurcations in two-dimensional, three-parameter discrete dynamical system with cubic terms[J]. AIMS Mathematics, 2026, 11(1): 1175-1201. doi: 10.3934/math.2026050

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Abstract

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.

References

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