Universality of Period Doubling in Coupled Maps

TL;DR

This paper investigates the universal properties of period doubling in coupled one-dimensional maps, revealing how the maximum order influences critical behavior through renormalization analysis and numerical validation.

Contribution

It identifies three fixed points in the renormalization operator and shows how eigenvalues depend on the maximum order for the zero-coupling fixed point, advancing understanding of universality in coupled maps.

Findings

Eigenvalues vary with maximum order for zero-coupling fixed point.

Renormalization results are confirmed by numerical methods.

Three fixed maps govern the critical behavior in coupled maps.

Abstract

We study the critical behavior of period doubling in two coupled one-dimensional maps with a single maximum of order $z$. In particurlar, the effect of the maximum-order $z$ on the critical behavior associated with coupling is investigated by a renormalization method. There exist three fixed maps of the period-doubling renormalization operator. For a fixed map associated with the critical behavior at the zero-coupling critical point, relevant eigenvalues associated with coupling perturbations vary depending on the order $z$, whereas they are independent of $z$ for the other two fixed maps. The renormalization results for the zero-coupling case are also confirmed by a direct numerical method.