Period-doubling bifurcations and routes to chaos of the vibratory systems contacting stops

A two-degree-of-freedom vibratory system contacting a single rigid stop is considered. The stop leads motions with repeat impacts, i.e., vibro-impacts. Such models play an important role in the studies of mechanical systems with amplitude constraining stops. Some non-typical routes to chaos, via period-doubling bifurcations of periodic motions with one impact, are studied. The period-doubling cascades of periodic orbits with one impact are usually discontinuous in the transition of periodic impact to chaos, in which there exist grazing bifurcation, torus bifurcation or Hopf bifurcation associated with multi-impact motions with higher period. A two-degree-of-freedom system having symmetrically placed rigid stops and subjected to periodic excitation is considered also. The system may exhibit more complicated dynamical behavior in the route to chaos via pitchfork bifurcation of period-one double-impact symmetrical motion. The routes to chaos, appearing in the vibro-impact systems, are qualitatively different form the typical period-doubling route to chaos in the usual consecutive maps.

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