Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

EXISTENCE OF PERIODIC SOLUTIONS FOR PLANAR HAMILTONIAN SYSTEMS AT RESONANCE

  • Received : 2010.03.15
  • Published : 2011.11.01
Download PDF
⟨ Previous Next ⟩

Abstract

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.

Keywords

References

  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.

Cited by

  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