Stability and transverse manifolds of periodic points for coupled maps

We consider the domains of stability of periodic points for linear-, internal- and external-coupled maps. We obtain the approximation expressions of the transverse manifolds at periodic points, which show that the transverse manifolds of periodic points are asymptotically elliptic paraboloids. We point out that the action of maps on the transverse manifold has the “symmetry” only in two-coupled maps. We study in detail the hereditary properties of domains of stability of period-doubling points for the internal-coupling of an arbitrary one-dimensional map with the help of its quadratic approximation and show that these domains follow the universal rules similar to the Feigenbaum universality rules for one-dimensional maps.