Two-parameter bifurcations of an impact system under different damping conditions. (September 2020) - Lyu, Xiaohong
Highlights: The periodic motion patterns and ( ω, δ )-parameter domains of an impact system are identified numerically. Different transition laws between neighboring p /1 ( p ≥ 0) motions through hysteresis and ligulate zones are analyzed for different cases of the damping ratio ζ . Codimension-2 bifurcation characteristics on the upper boundary of each ligulate zone are analyzed. Hysteresis phenomenon in the vicinity of period-doubling bifurcation is revealed. Abstract: 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.