This paper theoretically studies a simple system of two identical linear springs connected symmetrically to a mass in a V-shaped configuration, with an additional adjustable external force applied to the mass. As this force is varied, under certain conditions the equilibrium position of the mass demonstrates strong dependence on the history of changes in the external force, exhibiting hysteresis. Mathematically, variations of the external force cause the system to undergo two saddle-node bifurcations at two differing critical points, leading separately to the creation and destruction of branches of stable equilibria. Analysis of the bifurcation diagram shows that the saddle-node bifurcations cause hysteresis in the system, and the behavior is summarized in a hysteresis graph.

中文翻译：

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简单 V 形弹簧质量系统中的滞后

本文从理论上研究了一个简单的系统，该系统由两个相同的线性弹簧对称连接到一个 V 形配置的质量块，并在质量块上施加一个额外的可调外力。由于该力是变化的，在某些条件下，质量的平衡位置表现出对外力变化历史的强烈依赖，表现出滞后。在数学上，外力的变化导致系统在两个不同的临界点经历两个鞍点分叉，分别导致稳定平衡分支的创建和破坏。对分叉图的分析表明，鞍节点分叉会导致系统出现滞后现象，并将其行为总结在滞后图中。