QUASI-PERIODIC SADDLE-NODE BIFURCATIONS NEAR A DIFFERENTIABLE SINGULARITY FOR FORCED-OSCILLATIONS

[摘要] In the two-parameter unfolding of a Bogdanov-Takens singularity for autonomous differential equations in the plane with reflection symmetry, it is known in one case that there is a curve GAMMA in parameter space that corresponds to nonhyperbolic periodic orbits, and all one-parameter paths that cross GAMMA transversally give saddle-node bifurcations of periodic orbits. In the analogous situation for periodically forced systems, the curve GAMMA is replaced by a Cantor set GAMMA of parameter values that corresponds to nonhyperbolic quasi-periodic tori, and there is a restricted set of one-parameter paths that give quasi-periodic saddle-node bifurcations of tori. We require only finite differentiability of the system (C2 dependence on parameters, C(k) dependence on state variables, k greater-than-or-equal-to 29). The proof of this result uses a version of the Nash-Moser implicit function theorem that obtains C2 dependence of the implicitly defined function on parameters. (C) 1994 Academic Press, Inc.

[发布日期] 1994-06-01