PROPERTIES OF MODULATED ONE-DIMENSIONAL MAPS

Nonlinear recursive relations are examined under variation of a control parameter between two values A and B. Different periodic sequences (e.g. ABAB…, AABAB AABAB…, A⁶ B⁶ A⁶B⁶…) as well as random sequences are assumed. The graphical representation of the Lyapunov exponent on the A–B-plane yields an overwhelming richness of structures and allows the straightforward identification of several interesting features: multiple oscillatory states, self-similar periodic regions, closed and crossing superstable lines, order arising counterintuitively for processes which are chaotic for A alone and for B alone, and chaos at surprisingly low parameter values (”early chaos”) …