Tricritical Points and Bifurcations in a Quartic Map

Abstract

We have studied the 1-dimensional iteration map associated with the even quartic polynomial x n+l = 1 + a x 2 n + b x 4 n .This map allows a smooth transition from a single hump to a double hump. Bifurcations and higher-order transitions occur as we vary the parameters a, b. In addition to the usual universal bifurcation behavior discovered by Feigenbaum, we find a new universality class of bifurcations which is associated with a tricritical point. Tricritical points serve as natural boundaries toFeigenbaum critical lines. For the quartic map, the tricritical points which are the end-points of the original Feigenbaum line are (a,b) = (0, -1.59490) and (-2.81403, 1.40701). Associated with each tricritical point, there are two unstable directions as well as two independent exponents. The exponents are δ 1 T = 7.2851 andδ 2 T = 2.8571. At the tricritical point, we can introduce a universal function f* T (x) which obeys α T f* T (f* T (x/α T )) = f* T (x) with the scale factor α T = -1.69031. The quartic map has a, special duality transformation (a,b) ⇔ (a',ba'), such that the two mappings are intrinsically related. The tricritical points which are dual to the above pair of tricritical points are located at (-3.18980, 2.54371) and at (.95561, -1.14981) and are joined by a line which is the dual of the original Feigenbaum line. There are an infinite number of different tricritical points which form at least a Cantor set.