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Abstract

The iterative map xn+1 = rnxn„ (1-xn) is investigated with rn changing periodically between two values A and B. Different periodicities are assumed, e.g., {rn} = {BABA …} or {rn} = {BBABA BBABA…}. The Lyapunov exponent (a measure of average stability) is displayed with high resolution on the A-5-plane. The resulting images have aesthetically appealing self-similar structures. Furthermore, these images allow with one glimpse the identification of a number of system properties: coexistence of attractors, superstable curves, order by alternation of chaotic processes, and chaos by periodic resetting from a stable into an unstable fixed point. The chapter emphasizes that in contrast to most graphical representations of dynamical systems, the plane is defined by bifurcation parameters and not by phase variables. Thus, in the display the points do not wander on the plane as the dynamic process goes on. Instead, the dynamic process is calculated at each point {A, B} on the plane and the Lyapunov exponent is calculated as a time average over the iteration process. The Lyapunov exponent quantifies the average stability of the resulting oscillatory modes that may be periodic or chaotic.