Rich hidden dynamics in a two-parameter plane for spur gear system

As a key component of mechanical equipment, gear transmission exhibits intricate coexisting dynamics under the co-variation of parameters and initial values. Hidden dynamics can be dangerous, causing instability or system failure. Effectively unveiling hidden dynamics is crucial for gear systems and mechanical equipment. This study presents a method of computing coexistence dynamics in a two-parameter plane (TPP) considering multi-initial values based on the idea of gridding. The formation mechanism of coexistence dynamics in TPP is studied for a spur gear system. Diversity of the coexisting responses in TPP is clearly identified. Bifurcation details and evolutionary paths of the coexisting responses of the gear system are revealed through a designed bifurcation dendrogram. The results show that the superposition of existing regions of various dynamic responses in TPP creates diverse coexistence dynamics, such as coexistence of dynamic responses, coexistence of bifurcations, and coexistence of bifurcations and dynamic responses. Multi-bifurcation points and 23 coexisting responses of the gear system in TPP are observed. Two special bifurcation phenomena, incomplete bifurcation (IB) and complete branch bifurcation (CBB), of the coexisting responses under multi-initial values are initially obtained. IB and CBB not only alter the type of coexistence response, but also increase their quantity. IB induces the coexistence of dynamic responses and bifurcation, while CBB leads to two coexisting bifurcations (double bifurcations). All the dynamics of the gear system under the coordination of TPP and initial-value plane are fully revealed, providing valuable insights for the prediction and targeted control of the global dynamic behavior. The analysis method in this study can be extended to other nonlinear dynamic systems with time-varying parameters.