In this paper, three competitive systems with different kinds of state-dependent control are presented and investigated. The existence of the order-1 homoclinic orbit and order-1 periodic solution of the two systems that incorporate just one kind of state-dependent control is obtained by applying differential equation geometry theory, and the stability of the order-1 periodic solution of each system is also given. Besides, sufficient conditions for the existence and stability of the order-2 periodic solution of the system that incorporate two kinds of state-dependent control are gained by successor function method and analogue of Poincaré criterion, respectively. Finally, numerical simulations are carried out to verify the theoretical results.

中文翻译：

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具有状态反馈脉冲控制的两物种竞争系统的动力学分析

本文提出并研究了三种具有不同状态依赖控制的竞争系统。应用微分方程几何理论，得到了仅包含一种状态依赖控制的两个系统的一阶同宿轨道和一阶周期解的存在性，并且每个系统的一阶周期解的稳定性系统也给出。此外，分别通过后继函数法和庞加莱准则类比得到了包含两种状态相关控制的系统的二阶周期解存在和稳定的充分条件。最后通过数值模拟对理论结果进行验证。