Little is known about bifurcations in two-dimensional (2D) differential systems from the viewpoint of Kosambi–Cartan–Chern (KCC) theory. Based on the KCC geometric invariants, three types of static bifurcations in 2D differential systems, i.e. saddle-node bifurcation, transcritical bifurcation, and pitchfork bifurcation, are discussed in this paper. The dynamics far from fixed points of the systems generating bifurcations are characterized by the deviation curvature and nonlinear connection. In the nonequilibrium region, the nonlinear stability of systems is not simple but involves alternation between stability and instability, even though systems are invariably Jacobi-unstable. The results also indicate that the dynamics in the nonequilibrium region are node-like for three typical static bifurcations.

中文翻译：

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二维微分系统中三种分岔的几何不变量分析

从 Kosambi-Cartan-Chern (KCC) 理论的角度来看，对二维 (2D) 微分系统中的分岔知之甚少。基于KCC几何不变量，本文讨论了二维微分系统中的三种静态分岔，即鞍节点分岔、跨临界分岔和干草叉分岔。产生分岔的系统远离不动点的动力学特征是偏差曲率和非线性连接。在非平衡区域，系统的非线性稳定性并不简单，而是涉及稳定和不稳定之间的交替，即使系统总是雅可比不稳定的。结果还表明，对于三个典型的静态分岔，非平衡区域的动力学是节点状的。