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期刊论文

The Global Bifurcation Characteristics of the Forced van der Pol Oscillator

徐健学JIAN-XUE XU and JUN JIANG

chaos, Solitons & Fractals Vol. 7, No.1, pp. 3-19, 1996,-0001,():

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摘要/描述

In this paper, the bifurcation characteristics of the forced van der Pol oscillator on a specific parameter plane, including intermediate parameter regions, are investigated. The successive arrangement of the dominant mode-locking regions, where a single subharmonic solution with the rotation number, 1/(2k+1), exists, and the transitional zones between them are depicted. The transitional zones are explicitly proposed to be classified into two groups according to the different global characters:(1) the simple transitional zones, in which coexistence of two mode-locked solutions with rotation numbers 1/(2k+1) appear; (2) the complex transitional zones, in which the sub-zones with the mode-locked solutions, whose rotation numbers are rational fractions between 1/(2k+1) and 1/(2k-1), and the quasi-periodic solutions exist. The emphasis of this paper is to study the evolution of the global structures in the transitional zones. A complex transitional zone generally evolves from a Farey tree, when the forcing amplitude is small, to a chaotic regime, when forcing amplitude is sufficiently large. It is of great interest that the sub-zone with a rotation number, 1/2k, which has the largest width within a complex transitional zone, can usually intrude into the dominant regions of 1/(2k-1) before it completely vanishes. Moreover, the features of overlaps of mode-locking sub-zones and the number of coexistence of different attractors are also discussed.

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