# American Institute of Mathematical Sciences

April  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] J. Appell, V. Lakshmikantham, Nguyen Van Minh and P. P. Zabreiko, A general model of evolutionary processes. Exponential dichotomy - I, II,, Nonlinear Analysis TMA, 21 (1993), 207. doi: 10.1016/0362-546X(93)90111-5. [2] J. Appell, V. B. Moroz, A. Vignoli and P. P. Zabrejko, On the application of Kielhöfer's bifurcation theorem to Hammerstein equations with potential nonlinearity,, Boll. Unione Math. Ital., 8-B (1994), 833. [3] J. Appell and P. P. Zabrejko, Bifurcation points for integral equations of Hammerstein type,, J. Integral Equations Operator Theory, 16 (1993), 15. doi: 10.1007/BF01196600. [4] J. Awrejcewicz, M. Fečkan and P. Olejnik, On continuous approximation of discontinuous systems,, Nonlinear Analysis TMA, 62 (2005), 1317. doi: 10.1016/j.na.2005.04.033. [5] J. Awrejcewicz, M. Fečkan and P. Olejnik, Bifurcations of planar sliding homoclinics,, Mathematical Problems in Engineering, 2006 (2006), 1. doi: 10.1155/MPE/2006/85349. [6] F. Battelli and M. Fečkan, Chaos arising near a topologically transversal homoclinic set,, Topol. Methods Nonlinear Anal., 20 (2002), 195. [7] F. Battelli and M. Fečkan, On the chaotic behaviour of discontinuous systems,, J. Dynamics Differential Equations, 23 (2011), 495. doi: 10.1007/s10884-010-9197-7. [8] F. Battelli and M. Fečkan, Homoclinic trajectories in discontinuous systems,, J. Dynamics Differential Equations, 20 (2008), 337. doi: 10.1007/s10884-007-9087-9. [9] F. Battelli and M. Fečkan, An example of chaotic behaviour in presence of a sliding homoclinic orbit,, Annali di Matematica Pura ed Applicata, 189 (2010), 615. doi: 10.1007/s10231-010-0128-3. [10] F. Battelli and M. Fečkan, Bifurcation and chaos near sliding homoclinics,, J. Differential Equations, 248 (2010), 2227. doi: 10.1016/j.jde.2009.11.003. [11] F. Battelli and C. Lazzari, Exponential dichotomies, heteroclinic orbits, and Melnikov functions,, J. Differential Equations, 86 (1990), 342. doi: 10.1016/0022-0396(90)90034-M. [12] M. di Bernardo, C. J. Budd, A. R. Champneys and P. Kowalczyk, "Piecewise-smooth Dynamical Systems: Theory and Applications,", Springer, (2008). [13] B. Brogliato, "Nonsmooth Impact Mechanics,", Lecture Notes in Control and Information Sciences, (1996). [14] C. Chicone, "Ordinary Differential Equations with Applications,", Springer, (2006). [15] D. R. J. Chillingworth, Discontinuous geometry for an impact oscillator,, Dynamical Systems, 17 (2002), 389. doi: 10.1080/1468936021000041654. [16] S. N. Chow and J. K. Hale, "Methods of Bifurcation Theory,", Springer-Verlag, (1982). [17] Z. Du and W. Zhang, Melnikov method for homoclinic bifurcation in nonlinear impact oscillators,, Computers and Mathematics with Applications, 50 (2005), 445. doi: 10.1016/j.camwa.2005.03.007. [18] M. Fečkan, "Topological Degree Approach to Bifurcation Problems,", Springer, (2008). doi: 10.1007/978-1-4020-8724-0. [19] A. Fidlin, "Nonlinear Oscillations in Mechanical Engineering,", Springer, (2006). [20] J. Guckenheimer and P. Holmes, "Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields,", Springer-Verlag, (1983). [21] J. Knobloch, Lin's method for discrete dynamical systems,, J. Difference Equations and Applications, 6 (2000), 577. doi: 10.1080/10236190008808247. [22] A. Kovaleva, The Melnikov criterion of instability for random rocking dynamics of a rigid block with an attached secondary structure,, Nonlinear Analysis RWA, 11 (2010), 472. doi: 10.1016/j.nonrwa.2008.12.001. [23] M. A. Krasnosel'skii and P. P. Zabreiko, "Geometrical Methods of Nonlinear Analysis,", Springer-Verlag, (1984). doi: 10.1007/978-3-642-69409-7. [24] M. Kunze, "Non-smooth Dynamical Systems,", Springer, (2000). doi: 10.1007/BFb0103843. [25] R. I. Leine and H. Nijmeijer, "Dynamics and Bifurcations of Non-smooth Mechanical Systems,", Springer-Verlag, (2004). [26] S. Lenci and G. Rega, Heteroclinic bifurcations and optimal control in the nonlinear rocking dynamics of generic and slender rigid blocks,, Int. J. Bif. Chaos, 15 (2005), 1901. doi: 10.1142/S0218127405013046. [27] B. M. Levitan and V. V. Zhikov, "Almost Periodic Functions and Differential Equations,", Cambridge University Press, (1983). [28] X.-B. Lin, Using Melnikov's method to solve Silnikov's problems,, Proc. Roy. Soc. Edinburgh, 116A (1990), 295. doi: 10.1017/S0308210500031528. [29] O. Makarenkov and F. Verhulst, Bifurcation of asymptotically stable periodic solutions in nearly impact oscillators,, preprint, (). [30] K. R. Meyer and G. R. Sell, Melnikov transforms, Bernoulli bundles, and almost periodic perturbations,, Trans. Amer. Math. Soc., 314 (1989), 63. doi: 10.2307/2001437. [31] K. J. Palmer, Exponential dichotomies and transversal homoclinic points,, J. Differential Equations, 55 (1984), 225. doi: 10.1016/0022-0396(84)90082-2. [32] K. J. Palmer and D. Stoffer, Chaos in almost periodic systems,, Z. Angew. Math. Phys. (ZAMP), 40 (1989), 592. doi: 10.1007/BF00944809. [33] B. Sandstede, "Verzweigungstheorie Homokliner Verdopplungen,", PhD thesis, (1993). [34] D. Stoffer, Transversal homoclinic points and hyperbolic sets for non-autonomous maps I, II,, Z. angew. Math. Phys. (ZAMP), 39 (1988), 518. [35] A. Vanderbauwhede and B. Fiedler, Homoclinic period blow-up in reversible and conservative systems,, Z. Angew. Math. Phys. (ZAMP), 43 (1992), 292. doi: 10.1007/BF00946632. [36] P. P. Zabrejko and A. I. Povolockij, Bifurcation points of Hammerstein's equations,, Soviet Mathematics: Doklady, 11 (1970), 1220. [37] W. Xu, J. Feng and H. Rong, Melnikov's method for a general nonlinear vibro-impact oscillator,, Nonlinear Analysis TMA, 71 (2009), 418. doi: 10.1016/j.na.2008.10.120.

show all references

##### References:
 [1] J. Appell, V. Lakshmikantham, Nguyen Van Minh and P. P. Zabreiko, A general model of evolutionary processes. Exponential dichotomy - I, II,, Nonlinear Analysis TMA, 21 (1993), 207. doi: 10.1016/0362-546X(93)90111-5. [2] J. Appell, V. B. Moroz, A. Vignoli and P. P. Zabrejko, On the application of Kielhöfer's bifurcation theorem to Hammerstein equations with potential nonlinearity,, Boll. Unione Math. Ital., 8-B (1994), 833. [3] J. Appell and P. P. Zabrejko, Bifurcation points for integral equations of Hammerstein type,, J. Integral Equations Operator Theory, 16 (1993), 15. doi: 10.1007/BF01196600. [4] J. Awrejcewicz, M. Fečkan and P. Olejnik, On continuous approximation of discontinuous systems,, Nonlinear Analysis TMA, 62 (2005), 1317. doi: 10.1016/j.na.2005.04.033. [5] J. Awrejcewicz, M. Fečkan and P. Olejnik, Bifurcations of planar sliding homoclinics,, Mathematical Problems in Engineering, 2006 (2006), 1. doi: 10.1155/MPE/2006/85349. [6] F. Battelli and M. Fečkan, Chaos arising near a topologically transversal homoclinic set,, Topol. Methods Nonlinear Anal., 20 (2002), 195. [7] F. Battelli and M. Fečkan, On the chaotic behaviour of discontinuous systems,, J. Dynamics Differential Equations, 23 (2011), 495. doi: 10.1007/s10884-010-9197-7. [8] F. Battelli and M. Fečkan, Homoclinic trajectories in discontinuous systems,, J. Dynamics Differential Equations, 20 (2008), 337. doi: 10.1007/s10884-007-9087-9. [9] F. Battelli and M. Fečkan, An example of chaotic behaviour in presence of a sliding homoclinic orbit,, Annali di Matematica Pura ed Applicata, 189 (2010), 615. doi: 10.1007/s10231-010-0128-3. [10] F. Battelli and M. Fečkan, Bifurcation and chaos near sliding homoclinics,, J. Differential Equations, 248 (2010), 2227. doi: 10.1016/j.jde.2009.11.003. [11] F. Battelli and C. Lazzari, Exponential dichotomies, heteroclinic orbits, and Melnikov functions,, J. Differential Equations, 86 (1990), 342. doi: 10.1016/0022-0396(90)90034-M. [12] M. di Bernardo, C. J. Budd, A. R. Champneys and P. Kowalczyk, "Piecewise-smooth Dynamical Systems: Theory and Applications,", Springer, (2008). [13] B. Brogliato, "Nonsmooth Impact Mechanics,", Lecture Notes in Control and Information Sciences, (1996). [14] C. Chicone, "Ordinary Differential Equations with Applications,", Springer, (2006). [15] D. R. J. Chillingworth, Discontinuous geometry for an impact oscillator,, Dynamical Systems, 17 (2002), 389. doi: 10.1080/1468936021000041654. [16] S. N. Chow and J. K. Hale, "Methods of Bifurcation Theory,", Springer-Verlag, (1982). [17] Z. Du and W. Zhang, Melnikov method for homoclinic bifurcation in nonlinear impact oscillators,, Computers and Mathematics with Applications, 50 (2005), 445. doi: 10.1016/j.camwa.2005.03.007. [18] M. Fečkan, "Topological Degree Approach to Bifurcation Problems,", Springer, (2008). doi: 10.1007/978-1-4020-8724-0. [19] A. Fidlin, "Nonlinear Oscillations in Mechanical Engineering,", Springer, (2006). [20] J. Guckenheimer and P. Holmes, "Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields,", Springer-Verlag, (1983). [21] J. Knobloch, Lin's method for discrete dynamical systems,, J. Difference Equations and Applications, 6 (2000), 577. doi: 10.1080/10236190008808247. [22] A. Kovaleva, The Melnikov criterion of instability for random rocking dynamics of a rigid block with an attached secondary structure,, Nonlinear Analysis RWA, 11 (2010), 472. doi: 10.1016/j.nonrwa.2008.12.001. [23] M. A. Krasnosel'skii and P. P. Zabreiko, "Geometrical Methods of Nonlinear Analysis,", Springer-Verlag, (1984). doi: 10.1007/978-3-642-69409-7. [24] M. Kunze, "Non-smooth Dynamical Systems,", Springer, (2000). doi: 10.1007/BFb0103843. [25] R. I. Leine and H. Nijmeijer, "Dynamics and Bifurcations of Non-smooth Mechanical Systems,", Springer-Verlag, (2004). [26] S. Lenci and G. Rega, Heteroclinic bifurcations and optimal control in the nonlinear rocking dynamics of generic and slender rigid blocks,, Int. J. Bif. Chaos, 15 (2005), 1901. doi: 10.1142/S0218127405013046. [27] B. M. Levitan and V. V. Zhikov, "Almost Periodic Functions and Differential Equations,", Cambridge University Press, (1983). [28] X.-B. Lin, Using Melnikov's method to solve Silnikov's problems,, Proc. Roy. Soc. Edinburgh, 116A (1990), 295. doi: 10.1017/S0308210500031528. [29] O. Makarenkov and F. Verhulst, Bifurcation of asymptotically stable periodic solutions in nearly impact oscillators,, preprint, (). [30] K. R. Meyer and G. R. Sell, Melnikov transforms, Bernoulli bundles, and almost periodic perturbations,, Trans. Amer. Math. Soc., 314 (1989), 63. doi: 10.2307/2001437. [31] K. J. Palmer, Exponential dichotomies and transversal homoclinic points,, J. Differential Equations, 55 (1984), 225. doi: 10.1016/0022-0396(84)90082-2. [32] K. J. Palmer and D. Stoffer, Chaos in almost periodic systems,, Z. Angew. Math. Phys. (ZAMP), 40 (1989), 592. doi: 10.1007/BF00944809. [33] B. Sandstede, "Verzweigungstheorie Homokliner Verdopplungen,", PhD thesis, (1993). [34] D. Stoffer, Transversal homoclinic points and hyperbolic sets for non-autonomous maps I, II,, Z. angew. Math. Phys. (ZAMP), 39 (1988), 518. [35] A. Vanderbauwhede and B. Fiedler, Homoclinic period blow-up in reversible and conservative systems,, Z. Angew. Math. Phys. (ZAMP), 43 (1992), 292. doi: 10.1007/BF00946632. [36] P. P. Zabrejko and A. I. Povolockij, Bifurcation points of Hammerstein's equations,, Soviet Mathematics: Doklady, 11 (1970), 1220. [37] W. Xu, J. Feng and H. Rong, Melnikov's method for a general nonlinear vibro-impact oscillator,, Nonlinear Analysis TMA, 71 (2009), 418. doi: 10.1016/j.na.2008.10.120.
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