Chaos in forced impact systems

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

Chaos in forced impact systems

Flaviano Battelli ^1,2,3, and Michal Fe?kan ^4,5,6,

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

Abstract
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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.

Keywords: Bernoulli shift, chaotic behavior, impact systems..

Mathematics Subject Classification: Primary: 34C23, 34C37, 37G20; Secondary: 70K44, 70K5.

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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show all references

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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