Advanced Nonlinear Studies 13 (2013), 179–189
Periodic Bouncing Solutions for
( To Klaus Schmitt, with esteem )
Università degli Studi di Trieste
P.le Europa 1, Trieste, I-34127 Italy
SISSA - International School for Advanced Studies
Via Bonomea 265, Trieste, I-34136 Italy
Communicated by Steve Cantrell
We prove the existence of a periodic solution to a nonlinear impactoscillator, whose restor-
ing force has an asymptotically linear behavior. To
so-called impactoscillator (see, e.g., [ 1 , 3 , 25 , 33 ]) where a particle hits a wall attracted towards it by an elastic force. The existence of bouncing periodic solutions of such systems has been discussed, for example, in [ 3 , 17 , 28 , 30 , 31 , 32 , 33 , 37 ]. However, to the best of our knowledge, it seems that similar existence results on rotating periodic solutions with impact on spheres (or cylinders) of positive radius have not been presented yet. Let us now explain in detail what we mean by the term “bouncing solution”, borrowing the
, presence of the impact damper in these cases can arrive at elevation of the vibration amplitude and at irregular character of the system motion as was proved by the simulations. Communicated by Juan L.G. Guirao Acknowledgements This work was supported by The Ministry of Education, Youth and Sports from the National Programme of Sustainability (NPU II) project “IT4Innovations excellence in science - LQ1602”. The work was also supported by the Czech Science Foundation, Grant No. 15-06621S. References  D. R. J. Chillingworth. (2010), Dynamics of an impactoscillator near
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F. Peterka's paper consider dynamics of the impactoscillator.
Some of the next paper titles are: Numerical and
experimental investigations of nonsmooth
mechanical systems; Nonlinear dynamics of
Mechanical systems with discontinuities; Driven
nonlinear oscillators for modelling cardiac
phenomena; Chaos, synchronization and
bifurcations in a driven R-L-diode circuit; Control
of chaos for pendulum systems; Delay, nonlinear
oscillators and shimmying wheels
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This complexity of the behaviour is due to
internal, rather than external dynamics.
For autonomous, continuous time nonlinear
systems it has been reported that chaos cannot
occur when η = 1,2. Confirming this result
engineering systems such as driven and impactoscillators and beam system given in , have
at least three state variables. Furthermore the
models used for chaos studies; Chua's circuits
Γ4], Lorenz equations  and pressure
transducer  are all in third order forms. This
is not the case for
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