# American Institute of Mathematical Sciences

May  2013, 33(5): 2189-2209. doi: 10.3934/dcds.2013.33.2189

## Application of the subharmonic Melnikov method to piecewise-smooth systems

 1 Mathematics Division, Department of Information Engineering, Niigata University, 8050 Ikarashi 2-no-cho, Nishi-ku, Niigata 950-2181, Japan

Received  December 2011 Revised  July 2012 Published  December 2012

We extend a refined version of the subharmonic Melnikov method to piecewise-smooth systems and demonstrate the theory for bi- and trilinear oscillators. Fundamental results for approximating solutions of piecewise-smooth systems by those of smooth systems are given and used to obtain the main result. Special attention is paid to degenerate resonance behavior, and analytical results are illustrated by numerical ones.
Citation: Kazuyuki Yagasaki. Application of the subharmonic Melnikov method to piecewise-smooth systems. Discrete & Continuous Dynamical Systems, 2013, 33 (5) : 2189-2209. doi: 10.3934/dcds.2013.33.2189
##### References:
 [1] V. I. Arnold, "Mathematical Methods of Classical Mechanics," $2^{nd}$ edition, Springer-Verlag, New York, 1989.  Google Scholar [2] V. I. Babitsky and V. L. Krupenin, "Vibration of Strongly Nonlinear Discontinuous Systems," Springer-Verlag, Berlin, 2001. Google Scholar [3] A. Buică, J. Llibre and O. Makarenkov, Asymptotic stability of periodic solutions for nonsmooth differential equations with application to the nonsmooth van der Pol oscillator, SIAM J. Math. Anal., 40 (2009), 2478-2495. doi: 10.1137/070701091.  Google Scholar [4] T. K. Caughey, Sinusoidal excitation of a system with bilinear hysteresis, Trans. ASME, J. Appl. Mech., 27 (1960), 640-643.  Google Scholar [5] C. Chicone, Lyapunov-Schmidt reduction and Melnikov integrals for bifurcation of periodic solutions in coupled oscillators, J. Differential Equations, 112 (1994), 407-447. doi: 10.1006/jdeq.1994.1110.  Google Scholar [6] M. di Bernardo, C. J. Budd, A. R. Champneys and P. Kowalczyk, "Piecewise-Smooth Dynamical Systems: Theory and Applications," Springer-Verlag, London, 2008.  Google Scholar [7] E. Doedel, A. R. Champneys, T. F. Fairgrieve, Y. A. Kuznetsov, B. Sandstede and X. Wang, "AUTO97: Continuation and Bifurcation Software for Ordinary Differential Equations (with HomCont)," Concordia University, Montreal, 1997 (an upgraded version is available at http://cmvl.cs.concordia.ca/auto/. Google Scholar [8] J. Glover, A. C. Lazer and P. J. McKenna, Existence and stability of large scale nonlinear oscillations in suspension bridges, Z. Angew. Math. Phys., 40 (1989), 172-200. doi: 10.1007/BF00944997.  Google Scholar [9] I. V. Gorelyshev and A. I. Neishtadt, On the adiabatic theory of perturbations for systems with elastic reflections, J. Appl. Math. Mech. (PMM), 70 (2006), 4-7. doi: 10.1016/j.jappmathmech.2006.03.015.  Google Scholar [10] I. V. Gorelyshev and A. I. Neishtadt, Jump in adiabatic invariant at a transition between modes of motion for systems with impacts, Nonlinearity, 21 (2008), 661-676. doi: 10.1088/0951-7715/21/4/002.  Google Scholar [11] B. D. Greenspan and P. Holmes, Homoclinic orbits, subharmonics and global bifurcations in forced oscillations, in "Nonlinear Dynamics and Turbulence'' (eds. G. I. Barenblatt, G. Iooss and D. D. Joseph), Pitman, Boston, MA, (1983), 172-214.  Google Scholar [12] J. Guckenheimer and P. Holmes, "Nonlinear Oscillations, Dynamical Systems, and Bifurcations ofVector Fields," Springer-Verlag, New York, 1983.  Google Scholar [13] V. K. Melnikov, On the stability of the center for time-periodic perturbations, Trans. Moscow Math. Soc., 12 (1963), 1-56.  Google Scholar [14] J. A. Murdock, "Perturbations: Theory and Methods," John Wiley & Sons, New York, 1991.  Google Scholar [15] A. H. Nayfeh and D. T. Mook, "Nonlinear Oscillations," John Wiley & Sons, New York, 1979.  Google Scholar [16] S. Wiggins, "Introduction to Applied Nonlinear Dynamical Systems and Chaos," Springer-Verlag, New York, 1990.  Google Scholar [17] K. Yagasaki, The Melnikov theory for subharmonics and their bifurcations in forced oscillations, SIAM J. Appl. Math., 56 (1996), 1720-1765. doi: 10.1137/S0036139995281317.  Google Scholar [18] K. Yagasaki, Second-order averaging and Melnikov analyses for forced non-linear oscillators, J. Sound Vibration, 190 (1996), 587-609. doi: 10.1006/jsvi.1996.0080.  Google Scholar [19] K. Yagasaki, Periodic and homoclinic motions in forced, coupled oscillators, Nonlinear Dynam., 20 (1999), 319-359. doi: 10.1023/A:1008336402517.  Google Scholar [20] K. Yagasaki, Melnikov's method and codimension-two bifurcations in forced oscillations, J. Differential Equations, 185 (2002), 1-24. doi: 10.1006/jdeq.2002.4177.  Google Scholar [21] K. Yagasaki, Degenerate resonances in forced oscillators, Discrete Continuous Dynam. Systems - B, 3 (2003), 423-438. doi: 10.3934/dcdsb.2003.3.423.  Google Scholar [22] K. Yagasaki, Nonlinear dynamics of vibrating microcantilevers in tapping mode atomic force microscopy, Phys. Rev. B, 70 (2004), 245419. Google Scholar [23] K. Yagasaki, Bifurcations and chaos in vibrating microcantilevers of tapping mode atomic force microscopy, Int. J. Non-Linear Mech., 42 (2007), 658-672. Google Scholar

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##### References:
 [1] V. I. Arnold, "Mathematical Methods of Classical Mechanics," $2^{nd}$ edition, Springer-Verlag, New York, 1989.  Google Scholar [2] V. I. Babitsky and V. L. Krupenin, "Vibration of Strongly Nonlinear Discontinuous Systems," Springer-Verlag, Berlin, 2001. Google Scholar [3] A. Buică, J. Llibre and O. Makarenkov, Asymptotic stability of periodic solutions for nonsmooth differential equations with application to the nonsmooth van der Pol oscillator, SIAM J. Math. Anal., 40 (2009), 2478-2495. doi: 10.1137/070701091.  Google Scholar [4] T. K. Caughey, Sinusoidal excitation of a system with bilinear hysteresis, Trans. ASME, J. Appl. Mech., 27 (1960), 640-643.  Google Scholar [5] C. Chicone, Lyapunov-Schmidt reduction and Melnikov integrals for bifurcation of periodic solutions in coupled oscillators, J. Differential Equations, 112 (1994), 407-447. doi: 10.1006/jdeq.1994.1110.  Google Scholar [6] M. di Bernardo, C. J. Budd, A. R. Champneys and P. Kowalczyk, "Piecewise-Smooth Dynamical Systems: Theory and Applications," Springer-Verlag, London, 2008.  Google Scholar [7] E. Doedel, A. R. Champneys, T. F. Fairgrieve, Y. A. Kuznetsov, B. Sandstede and X. Wang, "AUTO97: Continuation and Bifurcation Software for Ordinary Differential Equations (with HomCont)," Concordia University, Montreal, 1997 (an upgraded version is available at http://cmvl.cs.concordia.ca/auto/. Google Scholar [8] J. Glover, A. C. Lazer and P. J. McKenna, Existence and stability of large scale nonlinear oscillations in suspension bridges, Z. Angew. Math. Phys., 40 (1989), 172-200. doi: 10.1007/BF00944997.  Google Scholar [9] I. V. Gorelyshev and A. I. Neishtadt, On the adiabatic theory of perturbations for systems with elastic reflections, J. Appl. Math. Mech. (PMM), 70 (2006), 4-7. doi: 10.1016/j.jappmathmech.2006.03.015.  Google Scholar [10] I. V. Gorelyshev and A. I. Neishtadt, Jump in adiabatic invariant at a transition between modes of motion for systems with impacts, Nonlinearity, 21 (2008), 661-676. doi: 10.1088/0951-7715/21/4/002.  Google Scholar [11] B. D. Greenspan and P. Holmes, Homoclinic orbits, subharmonics and global bifurcations in forced oscillations, in "Nonlinear Dynamics and Turbulence'' (eds. G. I. Barenblatt, G. Iooss and D. D. Joseph), Pitman, Boston, MA, (1983), 172-214.  Google Scholar [12] J. Guckenheimer and P. Holmes, "Nonlinear Oscillations, Dynamical Systems, and Bifurcations ofVector Fields," Springer-Verlag, New York, 1983.  Google Scholar [13] V. K. Melnikov, On the stability of the center for time-periodic perturbations, Trans. Moscow Math. Soc., 12 (1963), 1-56.  Google Scholar [14] J. A. Murdock, "Perturbations: Theory and Methods," John Wiley & Sons, New York, 1991.  Google Scholar [15] A. H. Nayfeh and D. T. Mook, "Nonlinear Oscillations," John Wiley & Sons, New York, 1979.  Google Scholar [16] S. Wiggins, "Introduction to Applied Nonlinear Dynamical Systems and Chaos," Springer-Verlag, New York, 1990.  Google Scholar [17] K. Yagasaki, The Melnikov theory for subharmonics and their bifurcations in forced oscillations, SIAM J. Appl. Math., 56 (1996), 1720-1765. doi: 10.1137/S0036139995281317.  Google Scholar [18] K. Yagasaki, Second-order averaging and Melnikov analyses for forced non-linear oscillators, J. Sound Vibration, 190 (1996), 587-609. doi: 10.1006/jsvi.1996.0080.  Google Scholar [19] K. Yagasaki, Periodic and homoclinic motions in forced, coupled oscillators, Nonlinear Dynam., 20 (1999), 319-359. doi: 10.1023/A:1008336402517.  Google Scholar [20] K. Yagasaki, Melnikov's method and codimension-two bifurcations in forced oscillations, J. Differential Equations, 185 (2002), 1-24. doi: 10.1006/jdeq.2002.4177.  Google Scholar [21] K. Yagasaki, Degenerate resonances in forced oscillators, Discrete Continuous Dynam. Systems - B, 3 (2003), 423-438. doi: 10.3934/dcdsb.2003.3.423.  Google Scholar [22] K. Yagasaki, Nonlinear dynamics of vibrating microcantilevers in tapping mode atomic force microscopy, Phys. Rev. B, 70 (2004), 245419. Google Scholar [23] K. Yagasaki, Bifurcations and chaos in vibrating microcantilevers of tapping mode atomic force microscopy, Int. J. Non-Linear Mech., 42 (2007), 658-672. Google Scholar
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