Various spontaneous oscillations and Hopf bifurcation have been observed in hair bundles of auditory hair cells, which play very important roles in the auditory function. In the present paper, the bifurcations and chaos of spontaneous oscillations of hair bundles are investigated in a theoretical model to explain the experimental observations. Firstly, the equivalent negative stiffness and symmetrical characteristic of the model are acquired. The model exhibits coexisting attractors symmetrical to each other or an attractor with symmetry by itself. The attractors include stable focus, stable periodic oscillations, and chaotic oscillations. Secondly, except for the well-known subcritical and supercritical Hopf bifurcations from the stable focus to period-1 limit cycle, the complex bifurcations of spontaneous oscillation patterns such as period-doubling bifurcation cascade to chaos and intermittency between periodic limit cycles and chaos, are observed. Various chaotic oscillations are distinguished. Lastly, a complex bifurcation process containing multiple modes of oscillations and bifurcations mentioned above is obtained, which provides the relationships between different spontaneous oscillation patterns. The results present not only the well-known Hopf bifurcation, but also the various spontaneous oscillations including periodic and chaotic patterns, which are consistent with the recent experimental results. The complex bifurcation process presents a global view of the nonlinear dynamics of complex spontaneous oscillations of hair bundles, which is very important for the auditory function.

中文翻译：

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听觉毛细胞中毛束自发振荡的分叉与混沌

在听觉毛细胞的毛束中观察到各种自发振荡和Hopf分叉，它们在听觉功能中起着非常重要的作用。在本文中，在一个理论模型中研究了发束自发振荡的分叉和混沌，以解释实验观察。首先，获得了模型的等效负刚度和对称特性。该模型展示了相互对称的共存吸引子或自身对称的吸引子。吸引子包括稳定的焦点、稳定的周期性振荡和混沌振荡。其次，除了众所周知的亚临界和超临界 Hopf 分岔从稳定焦点到第 1 周期极限环，观察到自发振荡模式的复杂分岔，例如倍周期分岔级联到混沌以及周期性极限环和混沌之间的间歇性。各种混沌振荡被区分。最后，得到了包含上述多个振荡模式和分岔的复杂分岔过程，它提供了不同自发振荡模式之间的关系。结果不仅呈现了著名的 Hopf 分岔，还呈现了包括周期性和混沌模式在内的各种自发振荡，这与最近的实验结果一致。复杂的分叉过程呈现了毛束复杂自发振荡的非线性动力学的全局视图，这对听觉功能非常重要。