Abstract. A family of local diffeomorphisms of R" 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 Z2 symmetry. Illustrative examples of the occurrence of the singly degenerate period doubling in the context of periodically forced planar oscillators are also presented. Key words. period doubling, bifurcation, bifurcation function, Lyapunov-Schmidt, Z2 symmetry AMS(MOS) subject classifications. 39, 15 1. Introduction. This paper describes the local bifurcations that take place when we perturb a diffeomorphism G0 of R" which has a fixed point with a single eigenvalue equal to -1. Since G0 has a nonhyperbolic fixed point, it is necessary to consider higher-order (nonlinear) terms in order to describe the phase portraits near the fixed point of the map G0 , both by itself and also under perturbation in a family G .. , µ. E Rk. When G0 is a map of R, any even-order term in its Taylor series expansion can be eliminated by a change of variables. This is a direct result of the normal forms theorem. After eliminating the constant and second-order terms, the linear coefficient will be -1 and the sign of the resulting coefficient of the third-order term will determine whether G0 will undergo a supercritical or subcritical period doubling (flip) bifurcation [Ar], [GH]. If the third-order coefficient should happen to be zero (a higher-order degeneracy), then the sign of the fifth-order term becomes important. Perturbations of
We note that the earlier result REF on higher order degeneracies in a period-doubling uses Z 2 equivariant singularity theory.
Period Doubling with Higher-Order Degeneracies
Siam Journal on Mathematical Analysis
Mathematics