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http://dx.doi.org/10.4134/JKMS.2011.48.6.1143

EXISTENCE OF PERIODIC SOLUTIONS FOR PLANAR HAMILTONIAN SYSTEMS AT RESONANCE  

Kim, Yong-In (Department of Mathematics University of Ulsan)
Publication Information
Journal of the Korean Mathematical Society / v.48, no.6, 2011 , pp. 1143-1152 More about this Journal
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
planar Hamilton system; resonance; periodic solution;
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1 R. Ortega, Asymmetric oscillators and twist mappings, J. London Math. Soc. (2) 53 (1996), no. 2, 325-342.   DOI
2 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.   DOI   ScienceOn
3 X. Yang, Unboundedness of solutions of planar Hamiltonian systems. Differential & difference equations and applications, 1167-1176, Hindawi Publ. Corp., New York, 2006.
4 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.   DOI   ScienceOn
5 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.   DOI
6 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.   DOI
7 T. Ding, Nonlinear oscillations at a point of resonance, Sci. Sinica Ser. A 25 (1982), no. 9, 918-931.
8 C. Fabry and A. Fonda, Nonlinear resonance in asymmetric oscillators, J. Differential Equations 147 (1998), no. 1, 58-78.
9 C. Fabry and J. Mawhin, Oscillations of a forced asymmetric oscillator at resonance, Nonlinearity 13 (2000), no. 3, 493-505.   DOI   ScienceOn
10 A. Fonda, Positively homogeneous Hamiltonian systems in the plane, J. Differential Equations 200 (2004), no. 1, 162-184.   DOI   ScienceOn
11 A. Fonda and J. Mawhin, Planar differential systems at resonance, Adv. Differential Equations 11 (2006), no. 10, 1111-1133.
12 N. G. Lloyd, Degree Theory, University Press, Cambridge, 1978.
13 J. Massera, The existence of periodic solutions of systems of differential equations, Duke Math. J. 17 (1950), 457-475.   DOI