Accumulation points in nonlinear parameter lattices

In 1963 Myrberg determined a period-doubling cascade of the quadratic map to accumulate at 1.401155189… As found later, the geometric way with which model parameters approach this value has universal behavior and several characteristic exponents associated with it. In the present paper we discuss the existence of an infinite number of points characterized by the simultaneous accumulation of two or more bifurcation cascades. We present an accurate numerical determination of the vertices of a doubly infinite nonlinear lattice which lead to a point of double accumulation. In addition, we discuss the number-theoretic nature of irrationalities characterizing vertices. Novel classes of universality with characteristic exponents are conjectured to exist near points of multiple accumulations.