Monotonicity in the damped pendulum type equations |
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Authors: | Pei-Lin Zhang Xin Ma Zhi-Long Peng Wen-Xin Qin |
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Institution: | (1) Department of Mathematics, Suzhou University, Suzhou, 215006, P.R.China;(2) Golden Audit College, Nanjing Audit University, Nanjing, 210029, P.R.China |
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Abstract: | In this paper we investigate the monotonicity in the pendulum type equations with position dependent damping. We show that
the system is strongly monotone under the overdamped condition. In the underdamped case, the Poincaré map PT is strongly monotone in a forward invariant region provided the average of the external force is large enough. Combining
the strong monotonicity with the dissipation property we show that the Poincaré map has in the cylindrical phase space an
invariant circle, on which PT is actually an orientation preserving circle homeomorphism. A series of consequences has then been obtained, including the
existence and uniqueness of the average velocity. Furthermore, we discuss the smoothness of this invariant curve and the ergodicity
of PT on this curve.
Supported by National Natural Science Foundation of China (10771155, 10571131) and Natural Science Foundation of Jiangsu Province
(BK 2006046). |
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Keywords: | Mathematics Subject Classification (2000)" target="_blank">Mathematics Subject Classification (2000) 34C12 34C15 34C25 34C60 34D45 37C65 |
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