Growth of random sequences

Abstract

We study the behaviour of two classes of random sequences. In these sequences each new element is generated by adding together two or more of the previous elements, or multiples of previous elements in the sequence. At least one of the elements is chosen randomly from a probability distribution of the previous elements. We find that a wide range of different types of behaviour emerge from linear to exponential growth and that the sequences exhibit a remarkably diverse phase space. Interestingly, new transitions in phase space are observed when the generating equations correspond to the backward difference equations. © 2006 Springer-Verlag Berlin Heidelberg.