Phononic frequency combs (PFCs) represent an emerging attractive nonlinear vibrational phenomenon characterized by equidistant spectral lines. Despite the extensive experimental studies, the complex nonlinear mixing nature of PFCs continues to present significant challenges in solving and investigating their complete dynamics, which is difficult to achieve by existing computational approaches. In this paper, the entire solution space within a representative PFC induced by a 1:2 internal resonance is elucidated by conducting continuation computations and numerical long-time integrations. The proposed continuation approach is achieved by integrating our developed semi-analytical residue-regulating homotopy method (RRHM) with a pseudo arc-length continuation technique. In this solution space, we unearth wide-range nonlinear evolutions including overlapping intervals between the periodic and quasi-periodic branches, abundant multivalued sub-intervals, cyclic-fold (CF) bifurcations, and torus-doubling (TD) routes to chaos. In addition, multiple coexistences of a chaotic attractor and a periodic orbit, a chaotic attractor and a quasi-periodic orbit, as well as a periodic orbit and three quasi-periodic orbits are identified. Furthermore, we meticulously dissect and distinguish non-smooth variations in PFC morphology, which manifest as multiple jumps in comb spacing as the excitation frequency is swept across. This study could serve as a general guide for a comprehensive exploration of PFC dynamics and can offer insights to inform and inspire related experimental studies.

中文翻译：

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声子频率梳内的强非线性混合演化


声子频率梳（PFC）代表了一种新兴的有吸引力的非线性振动现象，其特征是等距谱线。尽管进行了大量的实验研究，但 PFC 复杂的非线性混合特性仍然对解决和研究其完整动力学提出了重大挑战，这是现有计算方法难以实现的。在本文中，通过连续计算和数值长时积分阐明了由 1:2 内部谐振引起的代表性 PFC 内的整个解空间。所提出的连续方法是通过将我们开发的半解析留数调节同伦方法（RRHM）与伪弧长连续技术相结合来实现的。在这个解空间中，我们发现了广泛的非线性演化，包括周期性和准周期性分支之间的重叠区间、丰富的多值子区间、循环折叠（CF）分叉以及通向混沌的环面倍增（TD）路径。此外，还发现了一个混沌吸引子和一个周期轨道、一个混沌吸引子和一个准周期轨道、一个周期轨道和三个准周期轨道的多重共存。此外，我们仔细剖析和区分 PFC 形态的非平滑变化，这表现为当激励频率扫过时梳状间距的多次跳跃。这项研究可以作为全面探索 PFC 动力学的一般指南，并可以为相关实验研究提供信息和启发。