Controllability properties are studied for control-affine systems depending on a parameter $\alpha$ and with constrained control values. The uncontrolled systems in dimension two and three are subject to a homoclinic bifurcation. This generates two families of control sets depending on a parameter in the involved vector fields and the size of the control range. A new parameter $\beta$ given by a split function for the homoclinic bifurcation determines the behavior of these control sets. It is also shown that there are parameter regions where the uncontrolled equation has no periodic orbits, while the controlled systems have periodic solutions arbitrarily close to the homoclinic orbit.

中文翻译：

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同宿分岔附近的可控性

根据参数研究控制仿射系统的可控性 $\alpha$并具有约束控制值。维度 2 和维度 3 中的不受控制的系统受到同宿分岔的影响。这根据所涉及的向量场中的参数和控制范围的大小生成两个控制集族。新参数$\beta$由同宿分岔的分裂函数给出的决定了这些控制集的行为。还表明存在不受控方程没有周期轨道的参数区域，而受控系统具有任意接近同宿轨道的周期解。