Publication Date:
2016-01-01
Description:
The van der Pol-Mathieu-Duffing equation x ̈ + ( Ω 0 2 + h 1 cos Ω 1 t + h 2 cos Ω 2 t ) x − ( α − β x 2 ) x ̇ − h 3 x 3 = h 4 Ω 3 2 cos x cos Ω 3 t is considered in this paper, where α , β , h 1 , h 2 , h 3 , h 4 , Ω 1 , Ω 2 are small parameters, α , β 〉 0, the frequency Ω 3 is large compared to Ω 1 and Ω 2 , the above parameters are real. For ∀ α , β 〉 0, we use KAM (Kolmogorov-Arnold-Moser) theory to prove that the van der Pol-Mathieu-Duffing equation possesses quasi-periodic solutions for most of the parameters Ω 0 , Ω 1 , Ω 2 , Ω 3 , it verifies some phenomenon of Fahsi and Belhaq [Commun. Nonlinear Sci. 14 , 244-253 (2009)] and can be regarded as a extension of Abouhazim et al. [Nonlinear Dyn. 39 , 395-409 (2005)].
Print ISSN:
0022-2488
Electronic ISSN:
1089-7658
Topics:
Mathematics
,
Physics
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