Harmonic Solutions of Duffing Equation with Singularity via Time Map

Jing Xia, Suwen Zheng, Baohong Lv, Caihong Shan

Department of Fundamental Courses, Academy of Armored Force Engineering, Beijing, China.

DOI: 10.4236/am.2014.510146                

Abstract

This paper is devoted to the study of second-order Duffing equation with singularity at the origin, where  tends to positive infinity as , and the primitive function as . By applying the phase-plane analysis methods and Poincaré-Bohl theorem, we obtain the existence of harmonic solutions of the given equation under a kind of nonresonance condition for the time map.

Keywords

Harmonic Solutions, Duffing Equation, Singularity, Time Map, Poincaré-Bohl Theorem

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	Ding, T. (2004) Applications of Qualitative Methods of Ordinary Differential Equations. Higher Education Press, Beijing.
[2]	Fonda, A. (1993) Periodic Solutions of Scalar Second-Order Differential Equations with a Singularity. Académie Royale de Belgique. Classe des Sciences. Mémoires, 8-IV, 68-98.
[3]	Lazer, A.C. and Solimini, S. (1987) On Periodic Solutions of Nonlinear Differential Equations with Singularities. Proceedings of the American Mathematical Society, 99, 109-114. http://dx.doi.org/10.1090/S0002-9939-1987-0866438-7
[4]	Wang, Z. (2004) Periodic Solutions of the Second Order Differential Equations with Singularity. Nonlinear Analysis: Theory, Methods & Applications, 58, 319-331. http://dx.doi.org/10.1016/j.na.2004.05.006
[5]	Wang, Z., Xia, J. and Zheng, D. (2006) Periodic Solutions of Duffing Equations with Semi-Quadratic Potential and Singularity. Journal of Mathematical Analysis and Applications, 321, 273-285. http://dx.doi.org/10.1016/j.jmaa.2005.08.033
[6]	Xia, J. and Wang, Z. (2007) Existence and Multiplicity of Periodic Solutions for the Duffing Equation with Singularity. Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 137, 625-645.
[7]	Ding, T. and Zanolin, F. (1991) Time-Maps for the Solvability of Periodically Perturbed Nonlinear Duffing Equations. Nonlinear Analysis: Theory, Methods & Applications, 17, 635-653. http://dx.doi.org/10.1016/0362-546X(91)90111-D
[8]	Ding, T. and Zanolin, F. (1993) Subharmonic Solutions of Second Order Nonlinear Equations: A Time-Map Approach. Nonlinear Analysis: Theory, Methods & Applications, 20, 509-532. http://dx.doi.org/10.1016/0362-546X(93)90036-R
[9]	Qian, D. (1993) Times-Maps and Duffing Equations Crossing Resonance Points. Science in China Series A: Mathematics, 23, 471-479.
[10]	Capietto, A., Mawhin, J. and Zanolin, F. (1995) A Continuation Theorem for Periodic Boundary Value Problems with Oscillatory Nonlinearities. Nonlinear Differential Equations and Applications, 2, 133-163. http://dx.doi.org/10.1007/BF01295308
[11]	Lloyd, N.G. (1978) Degree Theory. University Press, Cambridge.