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Pushed traveling fronts in monostable equations with monotone delayed reaction
Application of the subharmonic Melnikov method to piecewisesmooth systems
1.  Mathematics Division, Department of Information Engineering, Niigata University, 8050 Ikarashi 2nocho, Nishiku, Niigata 9502181, Japan 
References:
[1] 
V. I. Arnold, "Mathematical Methods of Classical Mechanics," $2^{nd}$ edition, SpringerVerlag, New York, 1989. 
[2] 
V. I. Babitsky and V. L. Krupenin, "Vibration of Strongly Nonlinear Discontinuous Systems," SpringerVerlag, 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), 24782495. doi: 10.1137/070701091. 
[4] 
T. K. Caughey, Sinusoidal excitation of a system with bilinear hysteresis, Trans. ASME, J. Appl. Mech., 27 (1960), 640643. 
[5] 
C. Chicone, LyapunovSchmidt reduction and Melnikov integrals for bifurcation of periodic solutions in coupled oscillators, J. Differential Equations, 112 (1994), 407447. doi: 10.1006/jdeq.1994.1110. 
[6] 
M. di Bernardo, C. J. Budd, A. R. Champneys and P. Kowalczyk, "PiecewiseSmooth Dynamical Systems: Theory and Applications," SpringerVerlag, 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), 172200. 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), 47. 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), 661676. doi: 10.1088/09517715/21/4/002. 
[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), 172214. 
[12] 
J. Guckenheimer and P. Holmes, "Nonlinear Oscillations, Dynamical Systems, and Bifurcations ofVector Fields," SpringerVerlag, New York, 1983. 
[13] 
V. K. Melnikov, On the stability of the center for timeperiodic perturbations, Trans. Moscow Math. Soc., 12 (1963), 156. 
[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," SpringerVerlag, New York, 1990. 
[17] 
K. Yagasaki, The Melnikov theory for subharmonics and their bifurcations in forced oscillations, SIAM J. Appl. Math., 56 (1996), 17201765. doi: 10.1137/S0036139995281317. 
[18] 
K. Yagasaki, Secondorder averaging and Melnikov analyses for forced nonlinear oscillators, J. Sound Vibration, 190 (1996), 587609. doi: 10.1006/jsvi.1996.0080. 
[19] 
K. Yagasaki, Periodic and homoclinic motions in forced, coupled oscillators, Nonlinear Dynam., 20 (1999), 319359. doi: 10.1023/A:1008336402517. 
[20] 
K. Yagasaki, Melnikov's method and codimensiontwo bifurcations in forced oscillations, J. Differential Equations, 185 (2002), 124. doi: 10.1006/jdeq.2002.4177. 
[21] 
K. Yagasaki, Degenerate resonances in forced oscillators, Discrete Continuous Dynam. Systems  B, 3 (2003), 423438. 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. NonLinear Mech., 42 (2007), 658672. 
show all references
References:
[1] 
V. I. Arnold, "Mathematical Methods of Classical Mechanics," $2^{nd}$ edition, SpringerVerlag, New York, 1989. 
[2] 
V. I. Babitsky and V. L. Krupenin, "Vibration of Strongly Nonlinear Discontinuous Systems," SpringerVerlag, 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), 24782495. doi: 10.1137/070701091. 
[4] 
T. K. Caughey, Sinusoidal excitation of a system with bilinear hysteresis, Trans. ASME, J. Appl. Mech., 27 (1960), 640643. 
[5] 
C. Chicone, LyapunovSchmidt reduction and Melnikov integrals for bifurcation of periodic solutions in coupled oscillators, J. Differential Equations, 112 (1994), 407447. doi: 10.1006/jdeq.1994.1110. 
[6] 
M. di Bernardo, C. J. Budd, A. R. Champneys and P. Kowalczyk, "PiecewiseSmooth Dynamical Systems: Theory and Applications," SpringerVerlag, 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), 172200. 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), 47. 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), 661676. doi: 10.1088/09517715/21/4/002. 
[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), 172214. 
[12] 
J. Guckenheimer and P. Holmes, "Nonlinear Oscillations, Dynamical Systems, and Bifurcations ofVector Fields," SpringerVerlag, New York, 1983. 
[13] 
V. K. Melnikov, On the stability of the center for timeperiodic perturbations, Trans. Moscow Math. Soc., 12 (1963), 156. 
[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," SpringerVerlag, New York, 1990. 
[17] 
K. Yagasaki, The Melnikov theory for subharmonics and their bifurcations in forced oscillations, SIAM J. Appl. Math., 56 (1996), 17201765. doi: 10.1137/S0036139995281317. 
[18] 
K. Yagasaki, Secondorder averaging and Melnikov analyses for forced nonlinear oscillators, J. Sound Vibration, 190 (1996), 587609. doi: 10.1006/jsvi.1996.0080. 
[19] 
K. Yagasaki, Periodic and homoclinic motions in forced, coupled oscillators, Nonlinear Dynam., 20 (1999), 319359. doi: 10.1023/A:1008336402517. 
[20] 
K. Yagasaki, Melnikov's method and codimensiontwo bifurcations in forced oscillations, J. Differential Equations, 185 (2002), 124. doi: 10.1006/jdeq.2002.4177. 
[21] 
K. Yagasaki, Degenerate resonances in forced oscillators, Discrete Continuous Dynam. Systems  B, 3 (2003), 423438. 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. NonLinear Mech., 42 (2007), 658672. 
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