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 and 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.

[2]

V. I. Babitsky and V. L. Krupenin, "Vibration of Strongly Nonlinear Discontinuous Systems," Springer-Verlag, Berlin, 2001.

[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.

[4]

T. K. Caughey, Sinusoidal excitation of a system with bilinear hysteresis, Trans. ASME, J. Appl. Mech., 27 (1960), 640-643.

[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.

[6]

M. di Bernardo, C. J. Budd, A. R. Champneys and P. Kowalczyk, "Piecewise-Smooth Dynamical Systems: Theory and Applications," Springer-Verlag, London, 2008.

[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/.

[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.

[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.

[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.

[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.

[12]

J. Guckenheimer and P. Holmes, "Nonlinear Oscillations, Dynamical Systems, and Bifurcations ofVector Fields," Springer-Verlag, New York, 1983.

[13]

V. K. Melnikov, On the stability of the center for time-periodic perturbations, Trans. Moscow Math. Soc., 12 (1963), 1-56.

[14]

J. A. Murdock, "Perturbations: Theory and Methods," John Wiley & Sons, New York, 1991.

[15]

A. H. Nayfeh and D. T. Mook, "Nonlinear Oscillations," John Wiley & Sons, New York, 1979.

[16]

S. Wiggins, "Introduction to Applied Nonlinear Dynamical Systems and Chaos," Springer-Verlag, New York, 1990.

[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.

[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.

[19]

K. Yagasaki, Periodic and homoclinic motions in forced, coupled oscillators, Nonlinear Dynam., 20 (1999), 319-359. doi: 10.1023/A:1008336402517.

[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.

[21]

K. Yagasaki, Degenerate resonances in forced oscillators, Discrete Continuous Dynam. Systems - B, 3 (2003), 423-438. doi: 10.3934/dcdsb.2003.3.423.

[22]

K. Yagasaki, Nonlinear dynamics of vibrating microcantilevers in tapping mode atomic force microscopy, Phys. Rev. B, 70 (2004), 245419.

[23]

K. Yagasaki, Bifurcations and chaos in vibrating microcantilevers of tapping mode atomic force microscopy, Int. J. Non-Linear Mech., 42 (2007), 658-672.

show all references

References:
[1]

V. I. Arnold, "Mathematical Methods of Classical Mechanics," $2^{nd}$ edition, Springer-Verlag, New York, 1989.

[2]

V. I. Babitsky and V. L. Krupenin, "Vibration of Strongly Nonlinear Discontinuous Systems," Springer-Verlag, Berlin, 2001.

[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.

[4]

T. K. Caughey, Sinusoidal excitation of a system with bilinear hysteresis, Trans. ASME, J. Appl. Mech., 27 (1960), 640-643.

[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.

[6]

M. di Bernardo, C. J. Budd, A. R. Champneys and P. Kowalczyk, "Piecewise-Smooth Dynamical Systems: Theory and Applications," Springer-Verlag, London, 2008.

[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/.

[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.

[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.

[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.

[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.

[12]

J. Guckenheimer and P. Holmes, "Nonlinear Oscillations, Dynamical Systems, and Bifurcations ofVector Fields," Springer-Verlag, New York, 1983.

[13]

V. K. Melnikov, On the stability of the center for time-periodic perturbations, Trans. Moscow Math. Soc., 12 (1963), 1-56.

[14]

J. A. Murdock, "Perturbations: Theory and Methods," John Wiley & Sons, New York, 1991.

[15]

A. H. Nayfeh and D. T. Mook, "Nonlinear Oscillations," John Wiley & Sons, New York, 1979.

[16]

S. Wiggins, "Introduction to Applied Nonlinear Dynamical Systems and Chaos," Springer-Verlag, New York, 1990.

[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.

[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.

[19]

K. Yagasaki, Periodic and homoclinic motions in forced, coupled oscillators, Nonlinear Dynam., 20 (1999), 319-359. doi: 10.1023/A:1008336402517.

[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.

[21]

K. Yagasaki, Degenerate resonances in forced oscillators, Discrete Continuous Dynam. Systems - B, 3 (2003), 423-438. doi: 10.3934/dcdsb.2003.3.423.

[22]

K. Yagasaki, Nonlinear dynamics of vibrating microcantilevers in tapping mode atomic force microscopy, Phys. Rev. B, 70 (2004), 245419.

[23]

K. Yagasaki, Bifurcations and chaos in vibrating microcantilevers of tapping mode atomic force microscopy, Int. J. Non-Linear Mech., 42 (2007), 658-672.

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