The vibro-impact system found in many practicalsystems, as a typical non-smooth dynamical system is importantin engineering. Its studies include theoretical analysis, numericalsimulations, applied and experimental studies. By establishing thePoincar\'{e} map, using the center manifold theorem and the theory of normalforms, the periodic motion stability, bifurcation and chaos of the vibro-impactsystem can be investigated. The theory of bifurcation and chaos of map isone of the bases in studying vibro-impact systems. The smooth nonlinear dynamics canbe partly extended to vibro-impact systems, but the discontinuity caused byimpact will affect applicability and effectiveness of some methods.The theoretical studies and engineering applications of stability,bifurcations, chaos and singularity of vibro-impact systems are surveyedin this paper. Finally, the discussion centers around some unsolved problemsof bifurcations and chaos of vibro-impact systems, and the advance in thecorresponding discrete map dynamics. The future research trends aresuggested.
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