Inverted period-doubling sequences in four-dimensional reversible maps and solutions to the renormalization equations

Abstract

The existence of Feigenbaum-like sequences of inverted period doublings in four-dimensional reversible maps is pointed out. Starting with a pair of symmetrically coupled de Vogelaere maps and truncating at terms of degree 3, we solve numerically the renormalization equation in a 12-dimensional parameter space. In addition to the three known solutions obtained by previous workers with quadratic maps, which correspond to ``normal'' period-doubling sequences, interesting new solutions for the fixed map of the renormalization operator are found. A few possible physical implications are discussed.