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Advanced Nonlinear Studies 13 (2013), 179–189 Periodic Bouncing Solutions for Nonlinear Impact Oscillators ( To Klaus Schmitt, with esteem ) Alessandro Fonda Università degli Studi di Trieste P.le Europa 1, Trieste, I-34127 Italy e-mail: Andrea Sfecci SISSA - International School for Advanced Studies Via Bonomea 265, Trieste, I-34136 Italy e-mail: Communicated by Steve Cantrell Abstract We prove the existence of a periodic solution to a nonlinear impact oscillator, whose restor- ing force has an asymptotically linear behavior. To

so-called impact oscillator (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

, Singapore, 2007. [4] DI BERNARDO, M.— BUDD, C. J.— CHAMPNEYS, A. R.— KOWALCZYK, P.: Piecewise-smooth Dynamical Systems: Theory and Applications. Appl. Math. Sci. 163, Springer-Verlag, London, 2008. [5] BROGLIATO, B.: Nonsmooth Impact Mechanics. Lecture Notes in Control and Inform. Sci. 220, Springer, Berlin, 1996. [6] CHICONE, C.: Ordinary Differential Equations with Applications. Texts Appl. Math. 34, Springer, New York, 2006. [7] CHILLINGWORTH, D. R. J.: Discontinuous geometry for an impact oscillator, Dyn. Syst. 17 (2002), 389–420.

, 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 [1] D. R. J. Chillingworth. (2010), Dynamics of an impact oscillator near

methods of contact mechanics represents an appropriate way to evaluate the dynamical chain behaviour. F. Peterka's paper consider dynamics of the impact oscillator. 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

Bibliography [B] P. Boyland, Dual billiards, twist maps, and impact oscillators, Nonlinearity 9:1411–1438 (1996). [Be] A. Beardon, The Geometry of Discrete Groups, Graduate Texts in Mathematics 91, Springer, New York (1983). [BKS] T. Bedford, M. Keane, and C. Series, eds., Ergodic Theory, Sym- bolic Dynamics, and Hyperbolic Spaces, Oxford University Press, Oxford (1991). [DeB] N. E. J. De Bruijn, Algebraic theory of Penrose’s nonperiodic tilings, Nederl. Akad. Wentensch. Proc. 84:39–66 (1981). [Da] Davenport, The Higher Arithmetic: An Introduction to the Theory

complicated as to be effectively un-predictable. This complexity of the behaviour is due to internal, rather than external dynamics. 289 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 impact oscillators and beam system given in [3], have at least three state variables. Furthermore the models used for chaos studies; Chua's circuits Γ4], Lorenz equations [5] and pressure transducer [6] are all in third order forms. This is not the case for

. Gritli et al. [ 8 ] intend to control chaos in the impulse hybrid dynamics of a 1-DOF impact mechanical oscillator by master-slave controlled synchronization. The master-slave synchronization problem is reformulated as the stabilization of the synchronization error by means of a state-feedback controller. They show the effectiveness of the proposed method for the control of chaos by applying the designed control input to the chaotic impact oscillator. Zhai et al. [ 9 ] consider the synchronization for coupled nonlinear systems with disturbances in input and measured

with one Degree of Freedom: Part II - Resalts of Analogue Computer Modelling of the Motion, Acta Technica CSAV, 1974, No 5, pp 569-580 [20] Peterka F., 1996, Bifurcations and transition phenomena in an impact oscillator, Chaos, Solitons and Fractals, 7, 10, 1635-1647. [21] RaSkovic D., Mehanika - Dinamika (Dynamics), Naucna Knjiga, 1972. [22] RaSkovic D., Teorija oscilacija (Theory of oscillatins), Naucna Knjiga, 1952. [23] Stoker, J. J., (1950), Nonlinear Vibrations, Interscience Publish.

Sciences, New York; American Mathematical Society, Providence, RI, 2005. [29] Z. Opial, Sur les périodes des solutions de l’équation différentielle x′′ + g(x) = 0, (French) Ann. Polon. Math 10 (1961), 49–72. [30] H.-O. Peitgen and K. Schmitt, Global analysis of two-parameter elliptic eigenvalue problems, Trans. Amer. Math. Soc. 283 (1984), 57-95. [31] D. Qian and P.J. Torres, Periodic motions of linear impact oscillators via the successor map, SIAM J. Math. Anal. 36 (2005), 1707-1725. [32] P.H. Rabinowitz, Nonlinear Sturm-Liouville problems for second order ordinary