Two-parameter bifurcations of an impact system under different damping conditions

The periodic motion patterns and (ω, δ)-parameter domains of an impact system are identified by means of constructing two Poincaré maps. The influence of the damping ratio ζ on the transition characteristics between adjacent p/1 (p ≥ 0) motions is analyzed, and the correlations between impact velocities of 1/n (n ≥ 1) motions, occurrence domains and system damping parameters are discussed. The transition between adjacent fundamental motions is continuous and reversible for the case of large ζ. However, such transition is irreversible for the case of small ζ. In the cases of small ζ and very small ζ, there exist two types of transition zones, namely ligulate and hysteresis zones, which lie between the (ω, δ)-parameter domains of adjacent p/1 (p ≥ 0) motions. We give a detailed analysis of dynamics in the ligulate and hysteresis zones. As ζ is small enough, a subharmonic inclusions zone appears in the bottom region of p/1 fundamental motion. In the subharmonic inclusions zones, the system presents rich dynamic behaviors, including periodic bubble, period-doubling cascade, grazing and saddle-node bifurcations, as well as hysteresis phenomena and chaos. An additional discovery is that a narrow hysteresis interval exists near the period-doubling bifurcation and the period-doubling bifurcation boundary locates at different positions under different initial conditions.