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

J. Differential Equations, 86 (1990), 342-366. doi: 10.1016/0022-0396(90)90034-M.  Google Scholar

[12]

Springer, London, 2008.  Google Scholar

[13]

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]

Springer-Verlag, Berlin, 1984. doi: 10.1007/978-3-642-69409-7.  Google Scholar

[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]

Proc. Roy. Soc. Edinburgh, 116A (1990), 295-325. doi: 10.1017/S0308210500031528.  Google Scholar

[29]

preprint, arXiv:0909.4354v1. Google Scholar

[30]

Trans. Amer. Math. Soc., 314 (1989), 63-105. doi: 10.2307/2001437.  Google Scholar

[31]

J. Differential Equations, 55 (1984), 225-256. doi: 10.1016/0022-0396(84)90082-2.  Google Scholar

[32]

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

[33]

PhD thesis, University of Stuttgart, 1993. Google Scholar

[34]

Z. angew. Math. Phys. (ZAMP), 39 (1988), 518-549, 783-812. Google Scholar

[35]

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

[36]

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

[37]

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

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]

J. Differential Equations, 86 (1990), 342-366. doi: 10.1016/0022-0396(90)90034-M.  Google Scholar

[12]

Springer, London, 2008.  Google Scholar

[13]

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]

Springer-Verlag, Berlin, 1984. doi: 10.1007/978-3-642-69409-7.  Google Scholar

[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]

Proc. Roy. Soc. Edinburgh, 116A (1990), 295-325. doi: 10.1017/S0308210500031528.  Google Scholar

[29]

preprint, arXiv:0909.4354v1. Google Scholar

[30]

Trans. Amer. Math. Soc., 314 (1989), 63-105. doi: 10.2307/2001437.  Google Scholar

[31]

J. Differential Equations, 55 (1984), 225-256. doi: 10.1016/0022-0396(84)90082-2.  Google Scholar

[32]

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

[33]

PhD thesis, University of Stuttgart, 1993. Google Scholar

[34]

Z. angew. Math. Phys. (ZAMP), 39 (1988), 518-549, 783-812. Google Scholar

[35]

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

[36]

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

[37]

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

[1]

Francesca Alessio, Vittorio Coti Zelati, Piero Montecchiari. Chaotic behavior of rapidly oscillating Lagrangian systems. Discrete & Continuous Dynamical Systems, 2004, 10 (3) : 687-707. doi: 10.3934/dcds.2004.10.687

[2]

Matthew Nicol. Induced maps of hyperbolic Bernoulli systems. Discrete & Continuous Dynamical Systems, 2001, 7 (1) : 147-154. doi: 10.3934/dcds.2001.7.147

[3]

Yi-Chiuan Chen. Bernoulli shift for second order recurrence relations near the anti-integrable limit. Discrete & Continuous Dynamical Systems - B, 2005, 5 (3) : 587-598. doi: 10.3934/dcdsb.2005.5.587

[4]

P. Adda, J. L. Dimi, A. Iggidir, J. C. Kamgang, G. Sallet, J. J. Tewa. General models of host-parasite systems. Global analysis. Discrete & Continuous Dynamical Systems - B, 2007, 8 (1) : 1-17. doi: 10.3934/dcdsb.2007.8.1

[5]

Chao Wang, Dingbian Qian, Qihuai Liu. Impact oscillators of Hill's type with indefinite weight: Periodic and chaotic dynamics. Discrete & Continuous Dynamical Systems, 2016, 36 (4) : 2305-2328. doi: 10.3934/dcds.2016.36.2305

[6]

A.V. Borisov, A.A. Kilin, I.S. Mamaev. Reduction and chaotic behavior of point vortices on a plane and a sphere. Conference Publications, 2005, 2005 (Special) : 100-109. doi: 10.3934/proc.2005.2005.100

[7]

Sebastian Springer, Heikki Haario, Vladimir Shemyakin, Leonid Kalachev, Denis Shchepakin. Robust parameter estimation of chaotic systems. Inverse Problems & Imaging, 2019, 13 (6) : 1189-1212. doi: 10.3934/ipi.2019053

[8]

M. L. Bertotti, Sergey V. Bolotin. Chaotic trajectories for natural systems on a torus. Discrete & Continuous Dynamical Systems, 2003, 9 (5) : 1343-1357. doi: 10.3934/dcds.2003.9.1343

[9]

Sheng Zhu, Jinting Wang. Strategic behavior and optimal strategies in an M/G/1 queue with Bernoulli vacations. Journal of Industrial & Management Optimization, 2018, 14 (4) : 1297-1322. doi: 10.3934/jimo.2018008

[10]

Matteo Petrera, Yuri B. Suris. Geometry of the Kahan discretizations of planar quadratic Hamiltonian systems. Ⅱ. Systems with a linear Poisson tensor. Journal of Computational Dynamics, 2019, 6 (2) : 401-408. doi: 10.3934/jcd.2019020

[11]

Jacques Demongeot, Dan Istrate, Hajer Khlaifi, Lucile Mégret, Carla Taramasco, René Thomas. From conservative to dissipative non-linear differential systems. An application to the cardio-respiratory regulation. Discrete & Continuous Dynamical Systems - S, 2020, 13 (8) : 2121-2134. doi: 10.3934/dcdss.2020181

[12]

Denis de Carvalho Braga, Luis Fernando Mello, Carmen Rocşoreanu, Mihaela Sterpu. Lyapunov coefficients for non-symmetrically coupled identical dynamical systems. Application to coupled advertising models. Discrete & Continuous Dynamical Systems - B, 2009, 11 (3) : 785-803. doi: 10.3934/dcdsb.2009.11.785

[13]

Susanna Terracini, Juncheng Wei. DCDS-A Special Volume Qualitative properties of solutions of nonlinear elliptic equations and systems. Preface. Discrete & Continuous Dynamical Systems, 2014, 34 (6) : i-ii. doi: 10.3934/dcds.2014.34.6i

[14]

Anna Kostianko, Sergey Zelik. Inertial manifolds for 1D reaction-diffusion-advection systems. Part Ⅰ: Dirichlet and Neumann boundary conditions. Communications on Pure & Applied Analysis, 2017, 16 (6) : 2357-2376. doi: 10.3934/cpaa.2017116

[15]

Anna Kostianko, Sergey Zelik. Inertial manifolds for 1D reaction-diffusion-advection systems. Part Ⅱ: periodic boundary conditions. Communications on Pure & Applied Analysis, 2018, 17 (1) : 285-317. doi: 10.3934/cpaa.2018017

[16]

Farrukh Mukhamedov, Otabek Khakimov. Chaotic behavior of the P-adic Potts-Bethe mapping. Discrete & Continuous Dynamical Systems, 2018, 38 (1) : 231-245. doi: 10.3934/dcds.2018011

[17]

Fátima Drubi, Santiago Ibáñez, David Rivela. Chaotic behavior in the unfolding of Hopf-Bogdanov-Takens singularities. Discrete & Continuous Dynamical Systems - B, 2020, 25 (2) : 599-615. doi: 10.3934/dcdsb.2019256

[18]

Hamid Norouzi Nav, Mohammad Reza Jahed Motlagh, Ahmad Makui. Modeling and analyzing the chaotic behavior in supply chain networks: a control theoretic approach. Journal of Industrial & Management Optimization, 2018, 14 (3) : 1123-1141. doi: 10.3934/jimo.2018002

[19]

Samuel Bowong, Jean Luc Dimi. Adaptive synchronization of a class of uncertain chaotic systems. Discrete & Continuous Dynamical Systems - B, 2008, 9 (2) : 235-248. doi: 10.3934/dcdsb.2008.9.235

[20]

Noah H. Rhee, PaweŁ Góra, Majid Bani-Yaghoub. Predicting and estimating probability density functions of chaotic systems. Discrete & Continuous Dynamical Systems - B, 2019, 24 (1) : 297-319. doi: 10.3934/dcdsb.2017144

2020 Impact Factor: 2.425

Metrics

  • PDF downloads (55)
  • HTML views (0)
  • Cited by (6)

Other articles
by authors

[Back to Top]