Tristable system with negative stiffness has attracted extensive attention in the low frequency vibration isolation and vibration energy harvester. As a low frequency vibration isolator, it can achieve high static stiffness and low dynamic stiffness. As a vibration energy harvester, it had a wider bandwidth for resonance than the bistable one. The introduction of negative stiffness may induce subharmonic resonance and chaos in the tristable system. Chaos usually brings disorder to mechanical vibration system. Subharmonic resonance plays the negative effect on low frequency vibration isolation because they will transfer the high frequency energy of the system to the low frequency, but it is beneficial to broaden the working frequency band of vibration energy harvester. In this paper, the subharmonic bifurcation and chaos of a class of tristable system with negative stiffness are studied. The piecewise linearized systems are established to approximate the system with tristable potential. In order to conduct Melnikov analysis, the homoclinic-heteroclinic orbits and periodic orbits for the unperturbed piecewise linearized system are obtained respectively. The subharmonic Melnikov method for nonsmooth systems with four switched manifolds is developed. The thresholds for homoclinic-heteroclinic chaos and subharmonic resonance are derived by using non-smooth Melnikov method. It provides a theoretical support not only for design of the vibration energy harvester to obtain wider working frequency band, but also for design of the vibration isolation system to avoid high frequency energy transfer to low frequency. Moreover, the phenomena for infinite subharmonic bifurcations to chaos from odd order subharmonic orbit and the coexistence for chaotic and subharmonic attractors are revealed. The subharmonic Melnikov method with four switching manifolds developed in this paper lays a foundation for the subharmonic resonance analysis of other nonsmooth tristable systems.

负刚度三稳态系统在低频隔振和振动能量收集器中引起了广泛关注。作为低频隔振器，它可以实现高静刚度和低动刚度。作为一种振动能量收集器，它具有比双稳态更宽的共振带宽。负刚度的引入可能会在三稳态系统中引起次谐波共振和混沌。混乱通常会带来混乱机械振动系统。次谐波共振对低频隔振有负面影响，因为它们会将系统的高频能量转移到低频，但有利于拓宽振动能量收集器的工作频段。本文研究了一类负刚度三稳态系统的次谐波分岔和混沌问题。建立分段线性化系统以逼近具有三稳态势的系统。为了进行Melnikov分析，分别得到了无扰动分段线性化系统的同宿-异宿轨道和周期轨道。开发了具有四个切换流形的非光滑系统的次谐波 Melnikov 方法。利用非光滑Melnikov方法推导了同宿-异宿混沌和次谐波共振的阈值。它不仅为振动能量采集器的设计获得更宽的工作频带提供了理论支持，而且为隔振系统的设计避免了高频能量向低频转移提供了理论支持。此外，还揭示了从奇次谐波轨道无限次谐波分岔到混沌的现象以及混沌和次谐波吸引子并存的现象。本文开发的具有四个开关流形的次谐波Melnikov方法为其他非光滑三稳态系统的次谐波共振分析奠定了基础。它不仅为振动能量采集器的设计获得更宽的工作频带提供了理论支持，而且为隔振系统的设计避免了高频能量向低频转移提供了理论支持。此外，还揭示了从奇次谐波轨道无限次谐波分岔到混沌的现象以及混沌和次谐波吸引子并存的现象。本文开发的具有四个开关流形的次谐波Melnikov方法为其他非光滑三稳态系统的次谐波共振分析奠定了基础。它不仅为振动能量采集器的设计获得更宽的工作频带提供了理论支持，而且为隔振系统的设计避免了高频能量向低频转移提供了理论支持。此外，还揭示了从奇次谐波轨道无限次谐波分岔到混沌的现象以及混沌和次谐波吸引子并存的现象。本文开发的具有四个开关流形的次谐波Melnikov方法为其他非光滑三稳态系统的次谐波共振分析奠定了基础。揭示了从奇次谐波轨道无限次谐波分岔到混沌的现象以及混沌和次谐波吸引子的共存现象。本文开发的具有四个开关流形的次谐波Melnikov方法为其他非光滑三稳态系统的次谐波共振分析奠定了基础。揭示了从奇次谐波轨道无限次谐波分岔到混沌的现象以及混沌和次谐波吸引子的共存现象。本文开发的具有四个开关流形的次谐波Melnikov方法为其他非光滑三稳态系统的次谐波共振分析奠定了基础。