THE GLOBAL BIFURCATION CHARACTERISTICS OF THE FORCED VAN DER POL OSCILLATOR

In this paper, the bifurcation characteristics of the forced van der P ol oscillator on a specific parameter plane, including intermediate pa rameter regions, are investigated. The successive arrangement of the d ominant mode-locking regions, where a single subharmonic solution with the rotation number, 1/(2k + 1), exists, and the transitional zones b etween them are depicted. The transitional zones are explicitly propos ed 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 betwe en 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 str uctures in the transitional zones. A complex transitional zone general ly 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, whic h has the largest width within a complex transitional zone, can usuall y intrude into the dominant regions of 1/(2k - 1) before it completely vanishes, Moreover, the features of overlaps of mode-locking sub-zone s and the number of coexistence of different attractors are also discu ssed.