We consider the scalar delay differential equation $$ \dot{x}(t)=-x(t)+f_{K}(x(t-1)) $$ with a nondecreasing feedback function $f_{K}$ depending on a parameter $K$, and we verify that a saddle-node bifurcation of periodic orbits takes place as $K$ varies. The nonlinearity $f_{K}$ is chosen so that it has two unstable fixed points (hence the dynamical system has two unstable equilibria), and these fixed points remain bounded away from each other as $K$ changes. The generated periodic orbits are of large amplitude in the sense that they oscillate about both unstable fixed points of $f_{K}$.

中文翻译：

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延迟微分方程周期轨道的鞍点分岔

我们考虑具有非递减反馈函数 $f_{K}$ 的标量延迟微分方程 $$ \dot{x}(t)=-x(t)+f_{K}(x(t-1)) $$在参数 $K$ 上，我们验证了当 $K$ 变化时会发生周期轨道的鞍点分岔。选择非线性 $f_{K}$ 使其具有两个不稳定的不动点（因此动态系统具有两个不稳定的平衡点），并且随着 $K$ 的变化，这些固定点彼此之间保持有界。生成的周期轨道振幅很大，因为它们围绕 $f_{K}$ 的两个不稳定固定点振荡。