Harmonic Solutions of Duffing Equation with Singularity via Time Map
Affiliation(s)
Department of Fundamental Courses, Academy of Armored Force Engineering, Beijing, China.
Department of Fundamental Courses, Academy of Armored Force Engineering, Beijing, China.
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.
Cite this paper
Xia, J. , Zheng, S. , Lv, B. and Shan, C. (2014) Harmonic Solutions of Duffing Equation with Singularity via Time Map. Applied Mathematics, 5, 1528-1534. doi: 10.4236/am.2014.510146.
Xia, J. , Zheng, S. , Lv, B. and Shan, C. (2014) Harmonic Solutions of Duffing Equation with Singularity via Time Map. Applied Mathematics, 5, 1528-1534. doi: 10.4236/am.2014.510146.
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.
[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.