Eventually Vanished Solutions of a Forced Li閚ard System

ZHANG Yong-xin

doi:10.3879/j.issn.1000-0887.2009.10.013

Volume 30 Issue 10

Turn off MathJax

Article Contents

Article Navigation > Applied Mathematics and Mechanics > 2009 > 30(10): 1251-1260

ZHANG Yong-xin. Eventually Vanished Solutions of a Forced Li閚ard System[J]. Applied Mathematics and Mechanics, 2009, 30(10): 1251-1260. doi: 10.3879/j.issn.1000-0887.2009.10.013

Citation:

ZHANG Yong-xin. Eventually Vanished Solutions of a Forced Li閚ard System[J]. Applied Mathematics and Mechanics, 2009, 30(10): 1251-1260. doi: 10.3879/j.issn.1000-0887.2009.10.013

Citation:

ZHANG Yong-xin. Eventually Vanished Solutions of a Forced Li閚ard System[J]. Applied Mathematics and Mechanics, 2009, 30(10): 1251-1260. doi: 10.3879/j.issn.1000-0887.2009.10.013

PDF( 413 KB)

Eventually Vanished Solutions of a Forced Li閚ard System

doi: 10.3879/j.issn.1000-0887.2009.10.013

ZHANG Yong-xin

Department of Mathematics, Sichuan University, Chengdu 610064, P. R. China

Received Date: 2009-04-09
Rev Recd Date: 2009-08-19
Publish Date: 2009-10-15

Abstract

Abstract

Eventually vanished solutions, a special class of bounded solutions which tend to 0→±∞, of a Linard system with a tmie-dependent force were found. Not assuming it to be a small perturbation of a Hamiltonian system, the well-known Melnikov method could not be employed to determine the existence of eventually vanished solutions. A sequence of periodically forced systems was applied to approximate the considered system and their periodic solutions were found, where the difficulties caused by the non-Hamiltonian form were overcome by applying the Schaudercs fixed point theorem. The fact that the sequence of those periodic solutions has an accumulation gave the existence of an eventually vanished solution of the forced Linard system.
- eventually vanished,
- bounded solution,
- non-Hamiltonian,
- accumu lation

FullText(HTML)

References(26)

References

[1]	Hahan W. Stability of Motion[M]. Berlin-New York：Springer-Verlag,1967.
[2]	Yoshizawa T. Stability Theory by Liapunov′s Second Method[M]. Takyo：The Math Soc of Japan, 1996.
[3]	Hale J K. Ordinary Differential Equations [M].2nd ed. New York：Willey-Interscience，1980.
[4]	Buica A, Gasull A, Yang J. The third order Melnikov function of a quadratic center under quadratic perturbations [J]. J Math Anal Appl， 2007，331(1):443-454. doi: 10.1016/j.jmaa.2006.09.008
[5]	Champneys A, Lord G,Computation of homoclinic solutions to periodic orbits in a reducedwater-wave problem [J]. Physica D: Nonlinear Phenomena,1997,102(1/2):101-124.
[6]	Chow S N，Hale J K,Mallet-Paret J. An example of bifurcation to homoclinic orbits [J]. J Differential Equations,1980,37(3):351-371. doi: 10.1016/0022-0396(80)90104-7
[7]	Dumortier F, LI Cheng-zhi, ZHANG Zhi-fen.Unfolding of a quadratic integrable system with two centers and two unbounded heteroclinic loops [J]. J Differential Equations, 1997,139(1):146-193. doi: 10.1006/jdeq.1997.3285
[8]	LI Cheng-zhi, Rousseau C.A system with three limit cycles appearing in a Hopf bifurcation and dying in a homoclinic bifurcation: the cusp of order 4 [J]. J Differential Equations, 1989,79(1):132-167. doi: 10.1016/0022-0396(89)90117-4
[9]	ZHU Chang-rong, ZHANG Wei-nian. Computation of bifurcation manifolds of linearly independent homoclinic orbits [J]. J Differential Equations, 2008,245(7):1975-1994. doi: 10.1016/j.jde.2008.06.029
[10]	Mawhin J, Ward J. Periodic solutions of second order forced Liénard differential equations at resonance [J]. Arch Math, 1983,41(2):337-351. doi: 10.1007/BF01371406
[11]	Omari P, Villari G, Zanolin F. Periodic solutions of Linard differential equations with one-sided growth restriction [J]. J Differential equations, 1987,67(2):278-293. doi: 10.1016/0022-0396(87)90151-3
[12]	Franks J. Generalizations of the Poincaré-Birkhoff theorem [J]. Annals of Math,1988,128(1):139-151. doi: 10.2307/1971464
[13]	Jacobowitz H. Periodic solutions of 〖AKx¨〗+f(x,t)=0 via Poincaré-Birkhoff theorem [J]. J Differential Equations, 1976,20(1):37-52. doi: 10.1016/0022-0396(76)90094-2
[14]	Andronov A, Vitt E, Khaiken S. Theory of Oscillators [M]. Oxford：Pergamon Press, 1966.
[15]	Guckenheimer J, Holmes P. Nonlinear Oscillation, Dynamical Systems, and Bifurcations of Vector Fields [M]. New York：Springer,1983.
[16]	Rabinowitz P. Homoclinic orbits for a class of Hamiltonian systems [J]. Proc Roy Soc Edinburgh,1990, 114〖WTHZ〗A(1):33-38.
[17]	Ambrosetti A, Rabinowitz P. Dual variational methods in critical point theory and applications [J]. J Funct Anal, 1973,14(2):349-381. doi: 10.1016/0022-1236(73)90051-7
[18]	Chow S N, Hale J K. Methods of Bifurcation Theory [M]. New York, Springer：1982.
[19]	Carri〖AKa~〗o P, Miyagaki O.Existence of homoclinic solutions for a class of time-dependent Hamiltonian systems [J]. J Math Anal Appl, 1999,230(1):157-172. doi: 10.1006/jmaa.1998.6184
[20]	Szulkin A, Zou W. Homoclinic orbits for asymptotically linear Hamiltonian systems [J]. J Funct Anal, 2001, 187(1):25-41. doi: 10.1006/jfan.2001.3798
[21]	Izydorek M, Janczewska J. Homoclinic solutions for a class of the second order Hamiltonian systems [J]. J Differential Equations, 2005, 219(2):375-389. doi: 10.1016/j.jde.2005.06.029
[22]	Izydorek M, Janczewska J. Homoclinic solutions for nonautonomous second order Hamiltonian systems with a coercive potential [J]. J Math Anal Appl, 2007,335(2):1119-1127. doi: 10.1016/j.jmaa.2007.02.038
[23]	Tang X H, Xiao L. Homoclinic solutions for nonautonomous second-order Hamiltonian systems with a coercive potential [J]. J Math Anal Appl, 2009,351(2):586-594. doi: 10.1016/j.jmaa.2008.10.038
[24]	Tang X H, Xiao L. Homoclinic solutions for a class of second-order Hamiltonian systems [J]. J Math Anal Appl, 2009,354(2):539-549. doi: 10.1016/j.jmaa.2008.12.052
[25]	Zelati V, Ekeland I, Séré E. A variational approach to homoclinic orbits in Hamiltonian systems [J]. Math Ann, 1990,288(1):133-160. doi: 10.1007/BF01444526
[26]	Sansone G, Conti R. Nonlinear Differential Equations[M]. New York：Pergamon Press，1964.