Period Doubling with Higher-Order Degeneracies

Abstract

A family of local difleomorphisms of ${\bf R}^n$ can undergo a period doubling (flip) bifurcation as an eigenvalue of a fixed point passes through $ - 1$. This bifurcation is either supercritical or subcritical, depending on the sign of a coefficient determined by higher-order terms. If this coefficient is zero, the resulting bifurcation is “degenerate.” The period doubling bifurcation with a single higher-order degeneracy is treated, as well as the more general degenerate period doubling bifurcation where a fixed point has $ - 1$ eigenvalue and any number of higher-order degeneracies. The main procedure is a Lyapunov–Schmidt reduction: period-2 orbits are shown to be in one-to-one correspondence with roots of the reduced “bifurcation function,“ which has ${\bf Z}^2$ symmetry. Illustrative examples of the occurrence of the singly degenerate period doubling in the context of periodically forced planar oscillators are also presented.