Iterative properties of a one-dimensional quartic map: Critical lines and tricritical behavior

This paper studies the iterative properties of the two-parameter family of maps xₙ₊₁=1+axₙ²+bxₙ⁴. These maps can have either one or three extrema. Multiple extrema lead to a complex region of iterative stability and to new types of iterative behavior. In particular, the Feigenbaum critical line (locus of 2ⁿ-cycle accumulations) present for b≈0 terminates at two tricritical points T and T^' with coordinates (a,b)=(0,-1.59490) and (-2.81403,1.40701), respectively. There is also a "dual" critical line terminating in two additional tricritical points ̃ \~T=(-3.18980,2.54371) and ̃ \~T^'=(0.95561,-1.14981). Behavior near the tricritical points is controlled by a new fixed point fT* of the functional recursion relation fₙ₊₁(x)=[1fₙ(1)]f⁽²⁾(xfₙ(1)). This fixed point has two relevant directions and, therefore, behavior near a tricritical point depends on two new universal numbers δT⁽¹⁾=7.28469 and δT⁽²⁾=2.85713. Crossover and scaling behavior near the tricritical points are explored.