Alternating period-doubling cascades

作者：Aston Philip J^*; Bristow Neil

来源：Nonlinearity, 2013, 26(9): 2553-2576.

DOI：10.1088/0951-7715/26/9/2553

We consider period-doubling cascades in two-dimensional iterated maps. We define forward and backward period-doubling bifurcations, and use these concepts to describe an alternating period-doubling cascade in which forward and backward period-doubling bifurcations alternate. By tracking the eigenvalues of a typical map throughout such cascades we show that two-dimensional maps may give rise to two qualitatively different alternating period-doubling cascades. We apply renormalization theory to one class of alternating period-doubling cascades, and derive universal spatial scalings for such cascades from fixed points of the appropriate renormalization operator. We also derive universal parameter scalings for these cascades from the eigenvalues of the linearization of the renormalization operator, and provide the corresponding eigenfunctions. The theory is illustrated by an example.

• 出版日期2013-9