For general three-dimensional piecewise linear systems, some explicit sufficient conditions are achieved for the existence of a heteroclinic loop connecting a saddle-focus and a saddle with purely real eigenvalues. Furthermore, certain sufficient conditions are obtained for the existence and number of periodic orbits induced by the heteroclinic bifurcation, through close analysis of the fixed points of the parameterized Poincaré map constructed along the hereroclinic loop. It turns out that the number can be zero, one, finite number or countable infinity, as the case may be. Some sufficient conditions are also acquired that guarantee these periodic orbits to be periodic sinks or periodic saddle orbits, respectively, and the main results are illustrated lastly by some examples.

对于一般的三维分段线性系统，存在一些明确的充分条件，用于存在将鞍焦点和鞍与纯实特征值相连的异斜环。此外，通过对沿斜斜环构造的参数化庞加莱图的固定点进行仔细分析，可以获得由杂斜分叉引起的周期性轨道的存在和数量的某些充分条件。事实证明，视情况而定，该数字可以是零，一，有限数或可数的无穷大。还获得了一些足够的条件，以保证这些周期性轨道分别为周期性凹陷或周期性鞍形轨道，最后通过一些例子说明了主要结果。