대한수학회지 (Journal of the Korean Mathematical Society)

  • 제48권6호
  • /
  • Pages.1143-1152
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  • 2011
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  • 0304-9914(pISSN)
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  • 2234-3008(eISSN)
대한수학회 (Korean Mathematical Society)
권호 표지
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EXISTENCE OF PERIODIC SOLUTIONS FOR PLANAR HAMILTONIAN SYSTEMS AT RESONANCE

  • 투고 : 2010.03.15
  • 발행 : 2011.11.01
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초록

The existence of periodic solutions for the planar Hamiltonian systems with positively homogeneous Hamiltonian is discussed. The asymptotic expansion of the Poincar$\acute{e}$ map is calculated up to higher order and some sufficient conditions for the existence of periodic solutions are given in the case when the first order term of the Poincar$\acute{e}$ map is identically zero.

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참고문헌

  1. J. M. Alonso and R. Ortega, Roots of unity and unbounded motions of an asymmetric oscillator, J. Differential Equations 143 (1998), no. 1, 201-220. https://doi.org/10.1006/jdeq.1997.3367
  2. A. Capietto, W. Dambroslo, and Z. Wang, Coexistence of unbounded and periodic so-lutions to perturbed damped isochronous oscillators at resonance, Proc. Roy. Soc. Edin-burgh Sect. A 138 (2008), no. 1, 15-32. https://doi.org/10.1017/S030821050600062X
  3. A. Capietto and Z. Wang, Periodic solutions of Lienard equations with asymmetric nonlinearities at resonance, J. London Math. Soc. (2) 68 (2003), no. 1, 119-132. https://doi.org/10.1112/S0024610703004459
  4. T. Ding, Nonlinear oscillations at a point of resonance, Sci. Sinica Ser. A 25 (1982), no. 9, 918-931.
  5. C. Fabry and A. Fonda, Nonlinear resonance in asymmetric oscillators, J. Differential Equations 147 (1998), no. 1, 58-78.
  6. C. Fabry and J. Mawhin, Oscillations of a forced asymmetric oscillator at resonance, Nonlinearity 13 (2000), no. 3, 493-505. https://doi.org/10.1088/0951-7715/13/3/302
  7. A. Fonda, Positively homogeneous Hamiltonian systems in the plane, J. Differential Equations 200 (2004), no. 1, 162-184. https://doi.org/10.1016/j.jde.2004.02.001
  8. A. Fonda and J. Mawhin, Planar differential systems at resonance, Adv. Differential Equations 11 (2006), no. 10, 1111-1133.
  9. N. G. Lloyd, Degree Theory, University Press, Cambridge, 1978.
  10. J. Massera, The existence of periodic solutions of systems of differential equations, Duke Math. J. 17 (1950), 457-475. https://doi.org/10.1215/S0012-7094-50-01741-8
  11. R. Ortega, Asymmetric oscillators and twist mappings, J. London Math. Soc. (2) 53 (1996), no. 2, 325-342. https://doi.org/10.1112/jlms/53.2.325
  12. Z.Wang, Coexistence of unbounded solutions and periodic solutions of Lienard equations with asymmetric nonlinearities at resonance, Sci. China Ser. A 50 (2007), no. 8, 1205-1216. https://doi.org/10.1007/s11425-007-0070-z
  13. X. Yang, Unboundedness of solutions of planar Hamiltonian systems. Differential & difference equations and applications, 1167-1176, Hindawi Publ. Corp., New York, 2006.

피인용 문헌

  1. Non-periodic damped vibration systems with sublinear terms at infinity: Infinitely many homoclinic orbits vol.92, 2013, https://doi.org/10.1016/j.na.2013.07.018
  2. Ground state homoclinic orbits of damped vibration problems vol.2014, pp.1, 2014, https://doi.org/10.1186/1687-2770-2014-106
  3. Multiple Homoclinics for Nonperiodic Damped Systems with Superlinear Terms 2016, https://doi.org/10.1007/s40840-016-0396-1
  4. On homoclinic orbits for a class of damped vibration systems vol.2012, pp.1, 2012, https://doi.org/10.1186/1687-1847-2012-102
  5. Nonperiodic Damped Vibration Systems with Asymptotically Quadratic Terms at Infinity: Infinitely Many Homoclinic Orbits vol.2013, 2013, https://doi.org/10.1155/2013/937128
  6. Ground state homoclinic orbits of superquadratic damped vibration systems vol.2014, pp.1, 2014, https://doi.org/10.1186/1687-1847-2014-230