Abstract In this paper, we investigate multiple existence of homoclinic solutions for a periodically perturbed N-dimensional autonomous differential equation with a degenerate homoclinic solution of degeneracy degree d. Known results were obtained with a functional perturbation, which is regarded as an infinite-dimensional parameter. In this paper we consider a single parameter perturbation, a special form of former's functional perturbation, and prove that the single parameter is enough to unfold all possibilities of linearly independent homoclinic solutions bifurcated from the unperturbed degenerate homoclinic one, which actually improves the known results. Furthermore, we prove that those homoclinic solutions are all transversal, showing co-existence of multiple chaotic motions.

中文翻译：

---

退化同宿轨道单参数扰动引起的多重混沌

摘要 在本文中，我们研究了具有简并度为d 的退化同宿解的周期性摄动N 维自治微分方程同宿解的多重存在性。已知结果是通过函数扰动获得的，该函数被视为无限维参数。在本文中，我们考虑单参数扰动，前者的功能扰动的一种特殊形式，并证明单个参数足以展开线性无关同宿解从未受扰动的退化同宿解分叉的所有可能性，这实际上改进了已知结果。此外，我们证明了这些同宿解都是横向的，表明多个混沌运动并存。