We address the problem of understanding the dynamics around typical singular points of 3D piecewise smooth vector fields. A model Z₀ in 3D presenting a T-singularity is considered and a complete picture of its dynamics is obtained in the following way: (i) Z₀ has an invariant plane \(\pi_0\) filled up with periodic orbits (this means that the restriction \(Z_{0|\pi_0}\) is a center around the singularity); (ii) All trajectories of Z₀ converge to the surface \(\pi_0\); (iii) given an arbitrary integer \(k \geq 0\) then Z₀ can be approximated by \(\pi_{0}\)-invariant piecewise smooth vector fields \(Z_{\varepsilon}\) such that the restriction \(Z_{\varepsilon|\pi_0}\) has exactly k-hyperbolic limit cycles; (iv) the origin can be chosen as an asymptotic stable equilibrium of \(Z_{\varepsilon}\) when k = 0; and finally, (v) Z₀ has infinite codimension in the set of all 3D piecewise smooth vector fields.

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3D分段光滑向量场中典型奇点的吸引性，退化性和余维

我们解决了理解3D分段光滑向量场的典型奇异点周围动力学问题。考虑以3D表示T奇异性的模型Z ₀，并通过以下方式获得其动力学的完整图片：（i）Z ₀的不变平面\（\ pi_0 \）充满了周期性轨道（这表示限制\（Z_ {0 | \ pi_0} \）是围绕奇点的中心）；（ii）Z _0的所有轨迹都收敛到表面\（\ pi_0 \）；（iii）给定任意整数\（k \ geq 0 \）然后Z ₀可以用\（\ pi_ {0} \）-不变分段平滑向量字段\（Z _ {\ varepsilon} \）近似，使得限制\（Z _ {\ varepsilon | \ pi_0} \）恰好具有k-双曲极限周期 （iv）当k = 0时，可以选择原点作为\（Z _ {\ varepsilon} \）的渐近稳定平衡；最后，（v）Z ₀在所有3个D分段光滑向量场的集合中具有无限余维。