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Báo cáo hóa học: " Research Article Three Solutions for Forced Duffing-Type Equations with Damping Term"

Tuyển tập báo cáo các nghiên cứu khoa học quốc tế ngành hóa học dành cho các bạn yêu hóa học tham khảo đề tài: Research Article Three Solutions for Forced Duffing-Type Equations with Damping Term | Hindawi Publishing Corporation Boundary Value Problems Volume 2011 Article ID 736093 11 pages doi 10.1155 2011 736093 Research Article Three Solutions for Forced Duffing-Type Equations with Damping Term Yongkun Li and Tianwei Zhang Department of Mathematics Yunnan University Kunming Yunnan 650091 China Correspondence should be addressed to Yongkun Li yklie@ynu.edu.cn Received 16 December 2010 Revised 6 February 2011 Accepted 11 February 2011 Academic Editor Dumitru Motreanu Copyright 2011 Y. Li and T. Zhang. This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited. Using the variational principle of Ricceri and a local mountain pass lemma we study the existence of three distinct solutions for the following resonant Duffing-type equations with damping and perturbed term u t ơu t f t u t Yg t u t p t a.e. t e 0 w u 0 0 u w and without perturbed term u t ơu t f t u t p t a.e. t e 0 w u 0 0 u w . 1. Introduction In this paper we consider the following resonant Duffing-type equations with damping and perturbed term u ơu t f t u ty Ag t u tf p t a.e. t e 0 w . 1.1 u 0 0 u w where Ơ A e R f g 0 w X R R and p 0 w R are continuous. Letting A. 0 in problem 1.1 leads to u t ơu t f t u t p t a.e. t e 0 w 1.2 u 0 0 ufw which is a common Duffing-type equation without perturbation. The Duffing equation has been used to model the nonlinear dynamics of special types of mechanical and electrical systems. This differential equation has been named after the 2 Boundary Value Problems studies of Duffing in 1918 1 has a cubic nonlinearity and describes an oscillator. It is the simplest oscillator displaying catastrophic jumps of amplitude and phase when the frequency of the forcing term is taken as a gradually changing parameter. It has drawn extensive attention due to the richness of its chaotic behaviour with a variety of interesting .