The main purpose of this paper is to analyze the Hopf-like bifurcations in 3D piecewise linear systems. Such bifurcations are characterized by the birth of a piecewise smooth limit cycle that bifurcates from a singular point located at the discontinuity manifold. In particular, this paper concerns systems of the form \({\dot{x}}=Ax+b^{\pm }\) which are ubiquitous in control theory. For this class of systems, we show the occurrence of two distinct types of Hopf-like bifurcations, each of which gives rise to a crossing limit cycle (CLC). Conditions on the system parameters for the coexistence of two CLCs and the occurrence of a saddle-node bifurcation of these CLCs are provided. Furthermore, the local asymptotic stability of the pseudo-equilibrium point is analyzed and applications in discontinuous control systems are presented.

本文的主要目的是分析3D分段线性系统中类似Hopf的分叉。这种分叉的特征在于分段平滑极限循环的产生，该分段平滑极限循环从位于不连续歧管处的奇异点分叉。特别是，本文涉及\（{\ dot {x}} = Ax + b ^ {\ pm} \）形式的系统在控制理论中无处不在。对于此类系统，我们显示了两种不同类型的类似Hopf的分叉，每种分叉都会产生交叉极限环（CLC）。提供了两个CLC并存以及这些CLC发生鞍形节点分叉的系统参数的条件。此外，分析了伪平衡点的局部渐近稳定性，并提出了在不连续控制系统中的应用。