On the scaling properties of the period-increment scenario in dynamical systems

Two one-dimensional dynamical systems discrete in time are presented, where the variation of one parameter causes a sequence of global bifurcations; at each bifurcation the period increases by a constant value (period-increment scenario, usually denoted as a period-adding scenario). We determine all the bifurcation points and the scaling constants of the period-increment scenario analytically. A re-injection mechanism, leading to the period-increment scenario, is discussed. It will be shown, that in systems with more than one parameter the scaling constants can depend on the values of the parameters. (C) 2000 Elsevier Science Ltd. All rights reserved.