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-dou bling transition to chaos and examine to what extent the associated univers ality 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 tri critical points appear as the terminal points of Feigenbaum's critical curv es in the parameter plane. For the 2D map the same type of critical behavio r does not occur in a rigorous sense. It may be observed as a kind of inter mediate asymptotics, however, when one considers a finite number of period doublings. We refer to the associated points in the parameter plane as pseu do-tricritical. For the Ikeda map, we present estimates of the number of pe riod doublings, after which the departure from the tricritical universality becomes essential. (C) 2001 Elsevier Science B.V. All rights reserved.