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Neimark–Sacker bifurcation of a rational difference equation with delay

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  • He, Zhimin

Abstract

This paper is concerned with the existence of the Neimark–Sacker bifurcation of the following rational difference equation with delay xn+1=β+xnxn−m,n=0,1,2,…, where x−m,x−m+1,…,x0,β∈(0,∞)andm∈{2,3,…}⋅ It is shown that this equation undergoes a Neimark–Sacker bifurcation when the parameter β passes a critical value. Furthermore, based on the normal form theory and the computational algorithm developed by K. Murakami in K. Murakami(2002), the explicit algorithm for determining the direction and stability of the Neimark–Sacker bifurcation is derived. An explicit approximate expression of the invariant closed curve caused by Neimark–Sacker bifurcation is given. Some numerical simulations are presented to illustrate the analytical results found.

Suggested Citation

  • He, Zhimin, 2026. "Neimark–Sacker bifurcation of a rational difference equation with delay," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 240(C), pages 137-152.
    • Handle: RePEc:eee:matcom:v:240:y:2026:i:c:p:137-152
    DOI: 10.1016/j.matcom.2025.07.013
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    References listed on IDEAS

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    1. Peng, Mingshu, 2005. "Multiple bifurcations and periodic “bubbling” in a delay population model," Chaos, Solitons & Fractals, Elsevier, vol. 25(5), pages 1123-1130.
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