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3 Discontinuity Mappings 45 Fig. 6. 40 Hz. Closed circles (•)belong to the upward scan of the excitation amplitude and open circles ( ) for the downward scan, [225]. in the bifurcation diagram shown in Fig. 7 for the case of p4 ←→ p4/2 . It was reported that the additional impact occurred at the turning point that immediately preceded the primary impact. This turning point was the closest to the barrier. At bifurcation, the motion between these two impacts experienced significant changes. In terms of discontinuity mapping, De Weger et al [225] explicitly showed that the square-root also survives soft impacts with a soft barrier.

3 Discontinuity Mappings 33 invariant. The dynamics of a piecewise linear oscillator close to grazing has been experimentally examined by Ing et al [445]. The experimental results revealed the occurrence of higher periodic responses after grazing. The complex dynamics associated with high-dimensional mapping as applied to vibro-impact systems has been considered in several studies. For example, the four-dimensional Poincar´e map of the vibro-impact system was reduced to a two-dimensional normal form using a center manifold reduction and a normal form technique [1068].

According to Stronge [964], they defined what is known as the energetic coefficient of restitution, e∗ , which is obtained by equating the total work done by the normal force, FN , during restitution, Fn νdt, to the work done by the normal v<0 force during compression times, −e∗2 Fn νdt, where v is the velocity. 4 Border–Collision or C–Bifurcation Border bifurcation occurs when a fixed point or a periodic orbit of a piecewise smooth system crosses or collides one of the boundaries between different phase spaces as a system parameter varies in quasi–statically manner [742].

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