American Institute of Mathematical Sciences

Advanced
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

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 & Continuous Dynamical Systems - S, 2013, 6 (4) : 861-890. doi: 10.3934/dcdss.2013.6.861
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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.  Google Scholar

[2]

Boll. Unione Math. Ital., 8-B (1994), 833-850.  Google Scholar

[3]

J. Integral Equations Operator Theory, 16 (1993), 15-37. doi: 10.1007/BF01196600.  Google Scholar

[4]

Nonlinear Analysis TMA, 62 (2005), 1317-1331. doi: 10.1016/j.na.2005.04.033.  Google Scholar

[5]

Mathematical Problems in Engineering, 2006 (2006), 1-13. doi: 10.1155/MPE/2006/85349.  Google Scholar

[6]

Topol. Methods Nonlinear Anal., 20 (2002), 195-215.  Google Scholar

[7]

J. Dynamics Differential Equations, 23 (2011), 495-540. doi: 10.1007/s10884-010-9197-7.  Google Scholar

[8]

J. Dynamics Differential Equations, 20 (2008), 337-376. doi: 10.1007/s10884-007-9087-9.  Google Scholar

[9]

Annali di Matematica Pura ed Applicata, 189 (2010), 615-642. doi: 10.1007/s10231-010-0128-3.  Google Scholar

[10]

J. Differential Equations, 248 (2010), 2227-2262. doi: 10.1016/j.jde.2009.11.003.  Google Scholar

[11]

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[12]

Springer, London, 2008.  Google Scholar

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Lecture Notes in Control and Information Sciences, Springer, Berlin, 1996.  Google Scholar

[14]

Springer, New York, 2006.  Google Scholar

[15]

Dynamical Systems, 17 (2002), 389-420. doi: 10.1080/1468936021000041654.  Google Scholar

[16]

Springer-Verlag, New York, 1982.  Google Scholar

[17]

Computers and Mathematics with Applications, 50 (2005), 445-458. doi: 10.1016/j.camwa.2005.03.007.  Google Scholar

[18]

Springer, 2008. doi: 10.1007/978-1-4020-8724-0.  Google Scholar

[19]

Springer, Berlin, 2006. Google Scholar

[20]

Springer-Verlag, New York, 1983.  Google Scholar

[21]

J. Difference Equations and Applications, 6 (2000), 577-623. doi: 10.1080/10236190008808247.  Google Scholar

[22]

Nonlinear Analysis RWA, 11 (2010), 472-479. doi: 10.1016/j.nonrwa.2008.12.001.  Google Scholar

[23]

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[24]

Springer, Berlin, 2000. doi: 10.1007/BFb0103843.  Google Scholar

[25]

Springer-Verlag, Berlin, 2004.  Google Scholar

[26]

Int. J. Bif. Chaos, 15 (2005), 1901-1918. doi: 10.1142/S0218127405013046.  Google Scholar

[27]

Cambridge University Press, New York, 1983.  Google Scholar

[28]

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[29]

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[30]

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[32]

Z. Angew. Math. Phys. (ZAMP), 40 (1989), 592-602. doi: 10.1007/BF00944809.  Google Scholar

[33]

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[34]

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[35]

Z. Angew. Math. Phys. (ZAMP), 43 (1992), 292-318. doi: 10.1007/BF00946632.  Google Scholar

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Soviet Mathematics: Doklady, 11 (1970), 1220-1223. Google Scholar

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Nonlinear Analysis TMA, 71 (2009), 418-426. doi: 10.1016/j.na.2008.10.120.  Google Scholar

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