Structure change of stable and unstable manifolds in two-dimensional maps: Period-doubling bifurcation

Abstract

In dissipative orientation preserving two-dimensional maps, the tangled structure of stable and unstable manifolds of saddles born in the sequence period-doubling bifurcation is investigated by using geometrical method. Two theorems are obtained. The first theorem makes clear the existence of persistent transverse intersection between the stable and unstable manifolds over a large parameter interval. Using this theorem, the accumulation relations of the unstable manifolds are obtained. The second theorem specifies the type of the first asymptotic heteroclinic tangency, and gives the order relation of several critical values at which the first asymptotic heteroclinic tangency and the first homoclinic tangency occur. Numerical evidence supporting the theoretical result is also given.