A discrete-time Holling-Tanner model with ratio-dependent functional response is examined. We show that the system experiences a flip bifurcation and Neimark-Sacker bifurcation or both together at positive fixed point in the interior of ℝ+2$\mathbb {R}^{2}_{+}$ when one of the model parameter crosses its threshold value. We concentrate our attention to determine the existence conditions and direction of bifurcations via center manifold theory. To validate analytical results, numerical simulations are employed which include bifurcations, phase portraits, stable orbits, invariant closed circle, and attracting chaotic sets. In addition, the existence of chaos in the system is justified numerically by the sign of maximum Lyapunov exponents and fractal dimension. Finally, we control chaotic trajectories exists in the system by feedback control strategy.

中文翻译：

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离散时间比率相关的 Holling-Tanner 模型中的动力学和混沌控制

检查了具有依赖于比率的功能响应的离散时间 Holling-Tanner 模型。我们表明，当模型参数之一交叉时，系统在 ℝ+2$\mathbb {R}^{2}_{+}$ 内部的正不动点处经历翻转分岔和 Neimark-Sacker 分岔或两者同时发生其阈值。我们集中注意力通过中心流形理论确定分岔的存在条件和方向。为了验证分析结果，采用了数值模拟，包括分岔、相图、稳定轨道、不变闭圆和吸引混沌集。此外，系统中混沌的存在是由最大李雅普诺夫指数和分形维数的符号在数值上证明的。最后，我们通过反馈控制策略来控制系统中存在的混沌轨迹。