Bifurcations and chaos in a system with impacts

A vibro-impact system is considered. A body bounces on a flat horizontal surface of the vibro-bench. Hopf bifurcations of the vibro-impact system, in two kinds of strong resonance cases (λ03=1 and λ04=1), are investigated. A Poincaré map of the system is established. The period 1 single-impact motion of the system and its stability are studied by analytical methods, and Hopf bifurcation values and intersecting conditions of the system in strong resonance cases are determined. A center manifold theorem technique is applied to reduce the Poincaré map to a two-dimensional one, which is put into normal form by theory of normal forms. By the theory of Hopf and subharmonic bifurcations of fixed points in R2-strong resonance, dynamic behavior of the vibro-impact system near points of resonance is analyzed. In the resonance case of λ03=1, the system generally exhibits unstable period-3 three-impact motion; in the resonance case of λ04=1, the system can exhibit stable period-4 four-impact motion and quasi-periodic motion. The theoretical analyses are verified by numerical solutions. Routes to chaos, in two kinds of strong resonance cases considered, are obtained by numerical simulations. Quasi-periodic impacts in non-resonance and weak resonance cases, doubling-periodic bifurcations and chaotic motions are also stated briefly by numerical analyses.