The global bifurcation characteristics of the forced van der Pol oscillator

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, , 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 appear; (2) the complex transitional zones, in which the sub-zones with the mode-locked solutions, whose rotation numbers are rational fractions between , 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, , which has the largest width within a complex transitional zone, can usually intrude into the dominant regions of 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.