# American Institute of Mathematical Sciences

August  2013, 6(4): 861-890. doi: 10.3934/dcdss.2013.6.861

## Chaos in forced impact systems

 1 Dipartimento di Ingegneria Industriale e Scienze Matematiche 2 Marche Polytecnic University, Via Brecce Bianche 1 3 60131 Ancona 4 Department of Mathematical Analysis and Numerical Mathematics 5 Comenius University 6 Mlynsk dolina, 842 48 Bratislava

Received  October 2011 Revised  February 2012 Published  December 2012

We follow a functional analytic approach to study the problem of chaotic behaviour in time-perturbed impact systems whose unperturbed part has a piecewise continuous impact homoclinic solution that transversally enters the discontinuity manifold. We show that if a certain Melnikov function has a simple zero at some point, then the system has impact solutions that behave chaotically. Applications of this result to quasi periodic systems are also given.
Citation: Flaviano Battelli, Michal Fe?kan. Chaos in forced impact systems. Discrete and Continuous Dynamical Systems - S, 2013, 6 (4) : 861-890. doi: 10.3934/dcdss.2013.6.861
##### References:
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##### References:
 [1] Nonlinear Analysis TMA, 21 (1993), 207-218, 219-225. doi: 10.1016/0362-546X(93)90111-5. [2] Boll. Unione Math. Ital., 8-B (1994), 833-850. [3] J. Integral Equations Operator Theory, 16 (1993), 15-37. doi: 10.1007/BF01196600. [4] Nonlinear Analysis TMA, 62 (2005), 1317-1331. doi: 10.1016/j.na.2005.04.033. [5] Mathematical Problems in Engineering, 2006 (2006), 1-13. doi: 10.1155/MPE/2006/85349. [6] Topol. Methods Nonlinear Anal., 20 (2002), 195-215. [7] J. Dynamics Differential Equations, 23 (2011), 495-540. doi: 10.1007/s10884-010-9197-7. [8] J. Dynamics Differential Equations, 20 (2008), 337-376. doi: 10.1007/s10884-007-9087-9. [9] Annali di Matematica Pura ed Applicata, 189 (2010), 615-642. doi: 10.1007/s10231-010-0128-3. [10] J. Differential Equations, 248 (2010), 2227-2262. doi: 10.1016/j.jde.2009.11.003. [11] J. Differential Equations, 86 (1990), 342-366. doi: 10.1016/0022-0396(90)90034-M. [12] Springer, London, 2008. [13] Lecture Notes in Control and Information Sciences, Springer, Berlin, 1996. [14] Springer, New York, 2006. [15] Dynamical Systems, 17 (2002), 389-420. doi: 10.1080/1468936021000041654. [16] Springer-Verlag, New York, 1982. [17] Computers and Mathematics with Applications, 50 (2005), 445-458. doi: 10.1016/j.camwa.2005.03.007. [18] Springer, 2008. doi: 10.1007/978-1-4020-8724-0. [19] Springer, Berlin, 2006. [20] Springer-Verlag, New York, 1983. [21] J. Difference Equations and Applications, 6 (2000), 577-623. doi: 10.1080/10236190008808247. [22] Nonlinear Analysis RWA, 11 (2010), 472-479. doi: 10.1016/j.nonrwa.2008.12.001. [23] Springer-Verlag, Berlin, 1984. doi: 10.1007/978-3-642-69409-7. [24] Springer, Berlin, 2000. doi: 10.1007/BFb0103843. [25] Springer-Verlag, Berlin, 2004. [26] Int. J. Bif. Chaos, 15 (2005), 1901-1918. doi: 10.1142/S0218127405013046. [27] Cambridge University Press, New York, 1983. [28] Proc. Roy. Soc. Edinburgh, 116A (1990), 295-325. doi: 10.1017/S0308210500031528. [29] preprint, arXiv:0909.4354v1. [30] Trans. Amer. Math. Soc., 314 (1989), 63-105. doi: 10.2307/2001437. [31] J. Differential Equations, 55 (1984), 225-256. doi: 10.1016/0022-0396(84)90082-2. [32] Z. Angew. Math. Phys. (ZAMP), 40 (1989), 592-602. doi: 10.1007/BF00944809. [33] PhD thesis, University of Stuttgart, 1993. [34] Z. angew. Math. Phys. (ZAMP), 39 (1988), 518-549, 783-812. [35] Z. Angew. Math. Phys. (ZAMP), 43 (1992), 292-318. doi: 10.1007/BF00946632. [36] Soviet Mathematics: Doklady, 11 (1970), 1220-1223. [37] Nonlinear Analysis TMA, 71 (2009), 418-426. doi: 10.1016/j.na.2008.10.120.
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