Two-parameter analysis of the scaling behavior at the onset of chaos: tricritical and pseudo-tricritical points

We discuss the so-called tricritical points at the border of the period-doubling transition to chaos and examine to what extent the associated universality applies to 2D dissipative maps. As a concrete example, the Ikeda map is studied together with its 1D analog. For the approximate 1D map, the tricritical points appear as the terminal points of Feigenbaum's critical curves in the parameter plane. For the 2D map the same type of critical behavior does not occur in a rigorous sense. It may be observed as a kind of intermediate asymptotics, however, when one considers a finite number of period doublings. We refer to the associated points in the parameter plane as pseudo-tricritical. For the Ikeda map, we present estimates of the number of period doublings, after which the departure from the tricritical universality becomes essential.