## Analytical determination of bifurcations in an impact oscillator.(English)Zbl 0813.70012

The title problem is examined with a one-dimensional model with the coordinate $$x$$ described by $$\ddot x + d \dot x + x = \alpha \cos \omega t$$, $$x < a$$, $$\dot x \to -r\dot x$$, $$x = a$$, where $$d > 0$$ and $$0 < r \leq 1$$ ($$r$$ is the coefficient of restitution). Only those solutions are investigated for which $$x = a$$ for $$t_ i$$ and $$t_ i + 2\pi/\omega$$ and not for $$t_ j$$ characterized by $$t_ i < t_ j < t_ i + 2\pi/\omega$$, i.e. one impact for one forcing period.
Solutions are analytically constructed, validity of the above inequality must be checked numerically. The investigated impact oscillator can undergo conventional bifurcations, i.e. flips and saddle-nodes, but in this paper special attention is paid to the so-called grazing bifurcations, this occurs when the velocity of impact of the steady-state periodic orbit is zero. Two types of the grazing bifurcations are described. It is stated: by parameter values, where these two ones coincide, a bifurcation with codimension 2 occurs. It is possible to calculate by the methods of this paper cases in which one impact is taken for $$N$$ ($$N = 2,3,4,\dots$$) forcing periods. For some particular cases, very instructive numerically calculated figures are given.

### MSC:

 70K20 Stability for nonlinear problems in mechanics 70F35 Collision of rigid or pseudo-rigid bodies 37G99 Local and nonlocal bifurcation theory for dynamical systems

### Keywords:

grazing bifurcations; bifurcation with codimension 2
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